Numerical Simulation of a single emitter colloid thruster in
pure droplet cone-jet mode
by
Jorge Alejandro Carretero Benignos
Submitted to the Department of Mechanical Engineering in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2005
C Massachusetts Institute of Technology 2005. All rights reserved.
..............
Author .................................
Dep
f Mechanical Engineering
December 10, 2004
C ertified by .............................................................................
....... ..
Manuel Martinez-Sinchez
Professor, Aeronautics and Astronautics
esis Supervisor
Read by ......................................................
..............
w
...........
Ain Sonin
Professor, Mechanical Engineering
A
Committee Chair
Accepted by........................................................
allit Anand
Chair, Committee on Graduate Students
MASSACHUS
SI
OF TECHNOLOGY
MAY 1 5 2005
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Numerical Simulation of a single emitter colloid thruster in pure droplet
cone-jet mode
by
Jorge Alejandro Carretero Benignos
Submitted to the Department of Mechanical Engineering
on Dec 10, 2004, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
New Scientific missions call for emerging propulsion technologies capable of fine tuning a
satellite's relative position and cancelling small disturbances. One candidate technology that
holds promise for these type of missions are colloidal thrusters. These thrusters are electrostatic accelerators which do not rely on gas ionization (plasma), are intrinsically small,
and operate at low power levels, while having small plume divergence angles to avoid spacecraft (S/C) contamination problems. Colloid thrusters deliver low thrust (0.1 pN/emitter)
which can be multiplied many times over by integrating them in microfabricated arrays.
An important advantage is that with proper choice of propellant and operating regime their
specific impulse can be tailored from 500-7000 seconds making them viable candidates for
a multitude of mission profiles.
In the past many research groups have worked on developing colloid thrusters; however,
their basic physics are still not completely understood. For this reason, we have undertaken
the development of a numerical simulation of a colloid thruster to complement experimental
and analytical research in the area. The goal of this project has been to create a flexible numerical tool to compute single-emitter current, droplet size, velocity, electric field strengths
for a given geometry, fluid, flow rate, and voltage.
We have approached this issue from the numerical perspective by developing a simulation
of a colloid thruster operating in the single cone-jet pure droplet mode. Our numerical
simulation models the cone-jet transition region of the colloid jet; starting from the needle
up to the extractor grid, thus reproducing a typical electrospray experimental configuration.
The liquid is modelled as an incompressible viscous fluid with constant conductivity. Surface
charge relaxation effects are included, and the potential and electric fields are solved for.
The equations have been simplified by employing a slenderness approximation except for
the free surface boundary conditions where the terms have been kept exact, in order to
reproduce the Taylor cone shape correctly.
Simulation results are shown to compare well to experimental data for a variety of liquids,
and flow rates. In more detail, this thesis presents (a) results for a broader parametric range
than previously published materials; (b) careful comparison to available experimental data
and scalings proposed by several authors; (c) validation of several simple sub-models for
the cone and jet sections; (d) analysis of length scales in the cone-jet transition region;
(e) clarification of the role of relative permittivity for the range 7-110; (f) discussion of
electrode geometry and counter-electrode effects; (g) preliminary results for the onset of ion
evaporation.
2
Thesis Supervisor: Manuel Martinez-Sainchez
Title: Professor Aero & Astro, MIT
Thesis Supervisor: Akintunde Ibitayo Akinwande
Title: Professor EECS, MIT
Thesis Supervisor: Vlad Hruby
Title: President, Busek, Co.
Thesis Supervisor: Ain Sonin
Title: Chairman, Department of Mechanical Engineering
3
Acknowledgments
"Men wanted for hazardous journey. Small wages, bitter cold, long months of complete
darkness, constant danger, safe return doubtful. Honor and recognition in case of success."
Atributed to Sir Ernest Shackleton, Imperial 71ansantarcticExpedition, 1914
An adventure indeed, and what an adventure!!!
First, I would like to thank my advisor Professor Manuel Martinez-Sinchez.
He has
been the best advisor I could have ever hope to have. Not only is he an extraordinarily
gifted scientist but, perhaps more importantly a good honest man, who genuinely cares for
his students. Thanks for letting me be part of your research group and for being always
there when I needed help. Also many thanks are due to Professor Sonin, who throughout
these years has been a great example. To Professor Tayo Akinwande, thanks for all your
insights, help during this project, and for teaching me the finer details of Nigerian spanish.
A special word for Dr. Vlad Hruby who graciously agreed to be part of my committee at
the eleventh hour. Thanks Vlad!
Tatsuo once said that our office is the sunniest windowless office at MIT. He was definitely right.
Especial thanks are due to the people that have made this experience so
enjoyable: Mr. Blateau, for being a great friend and teaching me words in French that I
shouldn't have learned. Tatsuo Onishi, who put up with all my wookie noises and laughed
as hard as he could to all my bad jokes!. Paulo Lozano one of my best friends (the authentic
SPL man) with whom sharing the office has always been a pleasure and lots of fun. Shannon
"turtle " thanks for everything, you are one of my best friends and someone I could always
count on to cheer up the day, thanks so much. Good Old Mark Santi, the man himself
thanks for the great times we had here, at conferences, and in Europe... what a great trip.
Kay, thanks for the amazing time we had in France, and here in the office. It is good to
know someone who can catch every quote as obscure as it may be. Karen, thanks so much
for the great coffees we have had. Noah, Yassir, and Ollie, the coolest SPL outpost in SSL
land, thanks guys!. Finally, thanks to the rest of SPL. This is a great lab!!
Thanks to the Baty's, Kate and Gordon, who adopted me and with whom I have shared
so many good times. Thanks for everything!
My mexican friends. When you are so far away from home your friends become your
4
family. All I can say is that I have the best family I could ever hope to have. Rodrigo
Quintero, thanks for being my first friend here, and a great one at that. Ante, Marcos, Ana
Barbara, Julio, Ana, and Martin (not quite mexican, a bit on the swedish side!) thanks
guys I couldn't have made it through the dark years of my master's degree without your
support. To Raymundo, my best friend throughout this adventure, thanks I wouldn't have
made it without your help. To the crowd of the Ph.D. years and the tribe of the Jorge's:
Gina (que hace el mejor pastel de tres leches de la comarca), Jorge Feuchtwanger, Jorge
"El Oso" Vieyra, Jorge Ulises, Juan Gonz6lez, Yeri, Laia, Nuria and the rest of the media
lab gang.
To my friends of the triathlon club!, thanks for those early morning swims and runs.
Thanks Miriam for organizing everything, you are amazing. Also thanks to the core group,
Patricio, Brian, Ian, Dave, Leigh, Kim and Micha.
To all my friends in Aero&Astro, and specially to Dave Robertson and Fred Cote, friends
whith whom it was always a pleasure to hang out with.
My friends in Puebla, sorry I have been away so long. Thanks Edgar, Gabriel, Diana,
Yuria, Pliego, and Sam. Although, I usually see you guys only twice a year, I always have
a great time down there. Thanks for supporting me all this time and always believing I
could make it.
And finally to my family my mom Olga Alicia, my dad Jorge, and my brother Eduardo.
I would have never been able to even try this without all your support and love. Los quiero
muchisimo, y les estoy infinitamente agradecido por todo el carinio y amor que siempre me
han dado. No hay palabras para agradecerles, y sepan que siempre, donde quiera que ande
estin en mi coraz6n y pensamiento. Siempre cerca. Esta tesis se la dedico a mis abuelos
Herman, Olga Alicia, Alfonso y Evita. Y con mucho carifio a L. y G., pues esta tesis es
para ustedes.
Esta tesis se termino de imprimir el dia 3 de Febrero de 2005,
en los talleres Italgraff. Chabacano Num. 65, Local "A".
Mexico D.F.
La Edicion consta de pocos ejemplares
y sobrantes para reposici6n.
5
Contents
1 Introduction
2
1.1
Motivation
. . . . . . . . . . . . . . . . . .
17
1.2
Challenges . . . . . . . . . . . . . . . . . . .
17
1.3
Contributions . . . . . . . . . . . . . . . . .
18
1.4
Organization
19
. . . . .. . . . . . . . . . . .
Colloid thruster technology background
20
2.1
History of colloid thruster development . . .
20
2.1.1
Colloid thruster technology- 1960's .
20
2.1.2
Colloid thruster- 1990's- present day
23
2.2
2.3
3
16
History of numerical studies of free jets
. .
24
2.2.1
Non-charged free jets . . . . . . . . .
25
2.2.2
Charged free jet simulations . . . . .
29
2.2.3
Mixed ion-drop regime simulations .
31
Summary
. . . . . . . . . . . . . . . . . . .
33
Physics
34
3.1
Taylor cone . . . . . . . . . . . . . . . . . .
34
3.2
Starting potential . . . . . . . . . . . . . . .
38
3.3
Cone-jet physics
39
3.4
. . . . . . . . . . . . . . .
3.3.1
Cone-jet transition length scales
3.3.2
Viscous effects
. .
39
. . . . . . . . . . . .
43
3.3.3
Current scaling . . . . . . . . . . . .
44
3.3.4
Drop characteristics
. . . . . . . . .
47
Mixed ion -droplet regime . . . . . . . . . .
48
6
3.5
4
5
3.4.1
Ion emission mechanism . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.4.2
Space charge effects
. . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Cone-jet numerical model
54
4.1
Computational domain......
4.2
Free jet model..... ..
..............................
...................................
54
55
4.2.1
Fluid mass and momentum equations..................
56
4.2.2
Charge Conservation Equations .....
58
4.2.3
Free surface boundary conditions.....................
.....................
60
4.3
Electrostatic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.4
Field enhanced ion evaporation . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.5
Total current computation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.6
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
64
Numerical Solvers
5.1
Time dependent jet evolution . . . . . . . . . . . . . . . . . . . . . . . . . .
64
5.1.1
Time marching numerical solver
. . . . . . . . . . . . . . . . . . . .
65
5.1.2
Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.2
Electric potential calculation
. . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.3
Integration of solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.3.1
Gridding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.3.2
Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . .
74
5.3.3
Residue calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.4
6 Results for droplets
6.1
6.2
77
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.1.1
Reference experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.1.2
Numerical results and Comparison . . . . . . . . . . . . . . . . . . .
78
Sample results and parameter studies . . . . . . . . . . . . . . . . . . . . . .
79
6.2.1
Electrode distance effect . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.2.2
Analytical estimates compared to numerical results . . . . . . . . . .
84
7
6.3
7
8
6.2.3
Energy Conservation . . . . . . .
86
6.2.4
Cone-Jet transition length scales
89
6.2.5
Conductivity effects . . . . . . .
92
6.2.6
Dielectric permittivity effects . .
94
6.2.7
Voltage parameter study . . . . .
95
Summary . . . . . . . . . . . . . . . . .
98
99
Results for mixed ion-droplet regime
. . . . . . . . . . . . .
99
. . . . . . .
101
Summary . . . . . . . . . . . . . . . . .
107
7.1
Expected results
7.2
Modified solvation energies
7.3
108
Conclusions and recommendations
8.1
Conclusions ..................
........ ..... .......
108
8.2
Recommendations.............
. . . . . . . . . . . . . . . . . . ..
109
8.3
Future work ..................
....... ..... ...... ..
110
8
List of Figures
2-1
Colloid thruster operating in the single cone-jet mode. Notice the cone transition section, breakup and plume, Courtesy P. Lozano, [47] . . . . . . . . .
2-2
21
Three successive frames (in intervals of 1/64 s) showing a drop of water
(which appears bright) forming a Taylor cone surrounded by oil. The first
frame shows a comparison with the theoretical prediction of the cone angle;
the other frames show the drop in two phases of its oscillation around this
cone (Taylor, [62]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-3
"Soap film, microsecond exposures of successive stages (a) of jet formation,
(b),(c),(d) subsequent collapse." Reproduced from G. I. Taylor, [62]
2-4
. . . .
. . . . .
22
A) single cone-jet regime. B) Highly stressed regime (seen at higher voltages
than the single cone-jet regime) . . . . . . . . . . . . . . . . . . . . . . . . .
2-6
22
Taylor cones under various operating conditions. Liquid : Ethylene-Glycol,
conductivity K = 0.07 Si/m. Courtesy Mangoubi and Lozano [501
2-5
21
23
Breakup of a filament into satellite and subsatellite drops. As the viscosity
ratio 1 is decreased, the final state of the system has an increasingly complicated fractal structure (Tjahjadi, Stone, and Ottino, [63]). The pictures are,
from top to bottom, for 1 = 0.01, 0.067, 0.4, 1, and 2.8.
2-7
. . . . . . . . . . .
Comparison of experimental and numerical solutions with the lubrication
equations of Eggers and Dupont, [10].
The lines represent the profile at
various time distances and at the time of pinchoff. . . . . . . . . . . . . . .
2-8
26
Strobe photograph of the overall picture of an unstable jet.
27
The shutter
speed is 3 ps. On large scales, it appears as a whipping rope, with blob-like
varicose protrusions. From Hohman et al FIG. 11., [32]
9
. . . . . . . . . . .
30
3-1
Ethylene Glycol Liquid menisci. Left meniscus V=0 V, right meniscus V =
2.5KV with superposed spherical coordinate system, photos courtesy Paulo
Lozano, M IT. ..................................
3-2
. 34
Structure of the cone jet for Ethylene-Glycol, r7 = 2, Re=10.2, courtesy of
Fernandez de la Mora and Loscertales [131 . . . . . . . . . . . . . . . . . . .
35
3-3
Conical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3-4
Meniscus and capillary arrangement. L is distance between needle and collector, Lt is the needle length, and ho is the inner radius of the conducting
needle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-5
38
Flow rate vs current plot showing the onset of ion evaporation. I pure droplet
current (I Oc
Q1/ 2 ), Iot = I+ I, I, accumulated ion current. Courtesy Paulo
Lozano, [47]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3-6
Possible electrospray ion emission zones . . . . . . . . . . . . . . . . . . . .
48
3-7
ion desorption from jet surface, image potential
50
3-8
Space charge effect (.,(Eq. 3.63) on field fraction (E,/E*), Eq. 3.62. Space
. . . . . . . . . . . . . . .
charge is estimated based on formamide K = 2.1 Si/m, I = 400 nA,
40~
800 V, q/m ~ 350 x 10 3 C/Kg, and E, ~ 1 V/nm. (1) Refers to a jet radius
h = 10 nm and (2) to a jet radius h = 30 nm . . . . . . . . . . . . . . . . .
3-9
51
Space charge effect as given by equation 3.66. Notice the two limiting cases:
Child-Langmuir space charge limited case e, = 0, and weak space charge
case,
e0
-
oo and e, - oo
. . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4-1
Simulation setup with conducting needle and ground extractor electrode
4-2
Local spherical coordinate system used for the derivation of the fluid equations 56
4-3
Section of the liquid jet showing the Control Volume C.V. 1 used to de-
.
rive 4.12 and C.V. 2 used in the derivation of equation 4.11. . . . . . . . . .
5-1
69
Comparison of numerical solutions for a finite length needle- electrode arrangem ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
59
Depiction of the ring elements used to calculate the normal and tangential
electric fields on the surface of the jet. . . . . . . . . . . . . . . . . . . . . .
5-2
56
72
Effect on needle-ground potential distribution for increasing needle aspect
ratio (Lt/2ho) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
73
5-4
Jet radius vs z-axis for a typical simulation. Needle exit radius ho = 6pm,
total length of liquid domain L = 150pm.
The markers show the z-axis
.
position of the grid nodes, notice the higher density near the needle exit.
6-1
Dimensionless spray current C [Eq. 3.51] versus t
74
[Eq. 3.50] comparison
between experimental (shaded symbols) and numerical data(unshaded symbols). Experimental data from [13].Numerical data based on various liquids
(Table 6.1) and flow rates (77> 1). . . . . . . . . . . . . . . . . . . . . . . .
6-2
Comparison of experimental data from [13] and [4] to numerical results for
various dielectric permittivities,(E) versus non-dimensional slope f = C/77.
6-3
78
.
80
Simulation results for FM, K = 0.005 Si/m ho = 10 pm, L = 300 pm and
applied voltage V = 1150 V. Two flow rates shown 77 = 1.6 (dash-dot line)
and 77 = 2.2 (solid line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-4
81
Close up of the cone-jet transition of figure 6-3, K = 0.005 Si/m ho = 10 pm,
L = 300 pm and applied voltage V = 1150 V. Two flow rates shown q = 1.6
(dash-dot line) and 7 = 2.2 (solid line) . . . . . . . . . . . . . . . . . . . . .
6-5
Simulation results for FM1, ho = 6pm, 17 = 1.26.
82
Electrode distance is
L = 150pm, L = 300pm, and L = 600pm, with applied voltages V = 911V,
V = 1050V, and V = 1188V respectively.
6-6
. . . . . . . . . . . . . . . . . . .
Simulation results for FM1, ho = 6pm, q = 1.26.
Electrode distance is
L = 150pm, L = 300pm, and L = 600pm, with applied voltages V = 1190V.
6-7
83
83
Comparison of analytical Estimates for Electric fields and jet radius to simulation data. Liquid data: FM2 at 77 = 1.26, ho = 2pm, L = 150pm
. . . .
84
. . .
87
. . . . . . . .
88
6-8
Energy breakdown (percentages) vs q for Formamide, K = 0.01 Si/m
6-9
q vs Ohmic dissipation fraction (Va/V) for multiple liquids.
6-10 Plot of Ohmic dissipation for various liquids. x-axis estimate based on equation 6.12 vs simulation data . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6-11 Cone to jet transition radius (R.) vs proposed length scales. top r* (Eq.
3.16), center Rd (Eq. 3.19) , bottom Rs . . . . . . . . . . . . . . . . . . . .
90
6-12 Breakup radius for TBP (various conductivities) (Rb) vs proposed length
scales. top r* (Eq.
3.16), center Rd (Eq. 3.19) , bottom Rg. Experimental
data reproduced from Gamero and Hruby [22].
11
. . . . . . . . . . . . . . . .
91
6-13 Formamide data 77 = 1.6, ho = 10 pm, L = 300 tim and applied voltage
V = 1150 V.
Two conductivities shown:
Solid line K=0.001 Si/m, and
dash-dot line K=0.005 Si/m . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-14 Close up of the cone-jet transition of figure 6-13.
Formamide data 7
1.6, ho = 10 pm, L = 300 Mm and applied voltage V = 1150 V.
92
=
Two
conductivities shown: Solid line K=0.001 Si/m, and dash-dot line K=0.005
Si/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6-15 Formamide-1 (E = 111), and Octanol data (f = 10.34) . Both cases with
77 = 1.96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-16 Total current (I = Id + Ic)
94
vs Voltage for water, 7 = 1.55, ho = 10pLm, L =
300pIm, original starting voltage V = 1372 V. . . . . . . . . . . . . . . . . .
95
6-17 Convection (I.) and Conduction (Icd) current vs Voltage behavior for water.
,q= 1.55, ho = 10pim, and L = 300pim. Reference starting voltage 1372 V
6-18 Effect on jet shape at different voltages. Liquid: Water at q = 1.55, ho
96
=
10ptm, and L = 300pm. Reference starting voltage 1372 V . . . . . . . . . .
6-19 Experimental data from Gamero-Castaflo [521 for glycerol.
Needle ho
96
=
50pIm, L = 4mm and fixed flow rate Qo = 12.5nl/s). Notice the change from
single cone-jet mode to the multiple cone-jet mode (highly stressed regime)
7-1
Fig.
97
11 from Gamero [20]. Current vs non-dimensional flow rate for for-
mamide solutions of various conductivities. The arrow points to increasing
conductivity. The minimum is associated with onset of ion evaporation. F07
(K=0.71 Si/m) through F22 (K= 2.20 Si/m) . . . . . . . . . . . . . . . . .
7-2
Ion current vs flow rate. Formamide + NaI, dashed line corresponds to 77,
Figure 5.4.2, from Lozano, [47] . . . . . . . . . . . . . . . . . . . . . . . . .
7-3
100
100
Current vs non-dimensional flow rate plot for formamide (K=0.01 Si/m).
ho = 2pim. Two sets of results Go = 0.85 eV and 0.825eV, Iot is the total
emitted current, I is the accumulated ion current. FDM is the predicted
current for the pure droplet regime, Eq. 3.45. . . . . . . . . . . . . . . . . .
12
101
7-4
Top plot : Depiction of jet radius (h), and ion current per unit length [3.57]
(A) vs axial position. Bottom plot: Electric currents vs axial position. I, jet
convection current (4.21), IM jet conduction current (4.22), Ii accumulated
ion current (3.58), Itot total current . . . . . . . . . . . . . . . . . . . . . . .
7-5
Comparison of surface charge behavior for droplets and mixed-regime. Compare to pure droplet cases, 6-13 . . . . . . . . . . . . . . . . . . . . . . . . .
7-6
103
Normalized comparison of the ion current density (continuous line) and ohmic
heating distribution 0 = 27rK(Eth)2 /(1 + cos a), see section 6.2.3
7-7
102
. . . . .
104
Results for ion evaporation using Higuera's method. Results for FM2 (K =
0.01 Si/m), Go
=
0.8 eV, c = 0.85 for Na+ solvated ions. Plots show spatial
evolution relative to z-axis; left:jet radius, center: Normal external E-field,
right: evaporated ion current density. Arrows point towards lower flow rates
(Q
7-8
= 4,3,2) x 10- 1 1 M3 /s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
Results for ion evaporation using Higuera's method. Plots show spatial evolution relative to z-axis.
Left: ion current per unit length; right: current
plots: ion current I, convection current Iv, conduction current Icd, and
total current Itot = Ii + Iv + Id. Arrows point towards lower flow rates
(Q
= 4,3,2) x 10- 1 1 m 3 /s
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
106
List of Tables
3.1
Main current scalings, [13, 17]. Note that in the case of the F. de la Mora
scale the current scaling holds even for Re* >> 1, showing the same trends
6.1
as the IE and VE current scaling. . . . . . . . . . . . . . . . . . . . . . . .
46
Properties of liquids used in the simulations from [13]
79
14
. . . . . . . . . . . .
Alla en la fuente habia un chorrito
Se hacia grandote, se hacia chiquito
Estaba de mal humor el pobre chorrito tenia calor...
Over there in the fountain,
there is a little spurt It gets very big, it gets very small
And he was very mad poor little spurt! he was so hot . . .
El Chorrito Cancion popular mexicana (mexican folk song),
de Francisco Gabilondo Soler "Cri-Cri"
15
Chapter 1
Introduction
In the last few years a shift in satellite/probe missions has taken place. This change has been
motivated by new scientific, economic and administrative mission requirements. Changing
needs for scientific missions call for high precision propulsion systems capable of fine tuning a satellite's relative position and cancelling small disturbances.
From the economic
standpoint, these satellites are now designed to increase cost savings, reliability, system redundancy and survivability for any given mission. It is within this realm that new concepts
have emerged moving away from stand-alone satellites to constellations of miniature ones.
Examples of this approach are the Emerald, SPHERES, and LISA missions. An important
drive for these missions is the miniaturization of the individual satellites (1000 Kg) to the
scale of microsatellites (100 Kg) and nanosatellites (10 Kg). As mentioned before many of
these missions require very precise relative position control, for instance the LISA mission,
calls for three satellites flying in formation with relative spacings of 5 million Kilometers
controlled down to nanometers in order to measure gravitational waves. In these case, the
job of the propulsion system is to cancel any extraneous perturbations which would otherwise affect the measurements. For these missions the thruster will have to be compact, fuel
efficient, and will be most likely power limited. Electrical propulsion engines are ideal for
these time insensitive and power limited maneuvers as they work at relatively high specific
impulse (I,, > 500sec), [34].
One technology that holds promise for these type of missions is that of colloidal thrusters.
These thrusters are electrostatic accelerators which do not rely on gas ionization (plasma)
are intrinsically small, and operate at low power levels, while having small plume divergence
16
angles to avoid spacecraft (S/C) contamination problems.
1.1
Motivation
Electrosprays and colloid thrusters have been heavily studied for many years. From these
studies much of the basic physics of the phenomenon has been learned. However, the topic is
far from exhausted and is an active research area. The motivation for our work stems from
a need to obtain a more complete picture of the physics at work in the electrospray. The
aim of this thesis is to develop a numerical simulation of the cone-jet operating mode. The
goal of this project is to create a flexible numerical tool to compute single-emitter current,
droplet size, velocity, electric field strengths and spreading rate for a given geometry, fluid,
flow rate, and voltage. The wealth of information obtained from such a tool is expected
to complement and to some extent help us gain a more complete understanding of this
complex electrohydrodynamic behavior.
1.2
Challenges
Electrospray modelling is by itself a complex and fascinating subject.
The complexity
stems from the non-trivial interaction between the evolving non-linear fluid jet structure,
its electrical charges, and accompanying electric field. This coupling manifests itself in the
conical fluid structure from which the high speed jet emanates. It is that point, where the
cone transforms into a jet, the so called transition that lies at the core of this work. The
challenge is to model this transition region including the relevant fluid, and electrostatic
behavior. The transition from the cone to the jet section reduces the jet radius by an order
of a 100. This important length scale change makes for a stiff mathematical and numerical
problem.
From a physics perspective, this region bridges the gap between two dissimilar flow
regimes, i.e. from a low speed conduction current controlled cone to high speed convection
current controlled jet. A further complication is that being a free surface problem, the
shape of the cone and jet are part of the solution and is in turn affected by the coupling of
the flow physics and electrostatics. In other words it is a fascinating problem.
One important issue has is that it has been made a point to do so without relying on
asymptotic estimates and other approximations.
17
These approximations are very useful,
but their reliance on assumptions may make them true only for certain conditions and
or regimes. The hope is that our work will help to confirm existing analytical estimates,
and asymptotic expressions, while pointing out some of their limitations, thus serving as a
stepping ground for other studies.
1.3
Contributions
Other numerical studies of this phenomenon have been published in the past. These works
although they have targeted the same basic phenomenon, have focused for the most part
on other operating regimes and lower conductivity liquids.
The contributions of this work are as follows :
1. A broader parametric range than previously published materials,
2. Careful comparison to available experimental data including
f(E),
I
Vs
V,and I vs Q
3. Validation of several simple sub-models, for the cone and jet section
4. Analysis of length scales in the transition region
5. Analysis of energy usage by the cone-jet.
6. Study of electrode spacing effects and needle boundary conditions
7. Integration of an ion evaporation model used to explore the mixed regime
In more detail, we have successfully compared our results to existing current scalings by
Fernindez de la Mora and Loscertales, as well as those by Gafin-Calvo. Simulations have
been carried for many liquids spanning over an order of magnitude variations in surface
tension, viscosity, relative permittivity and conductivity. Our results suggest, as originally
predicted by Gafain-Calvo and other authors, that the liquid relative permittivity may not
play a role in current scalings.
We have carried out various parametric studies with our simulation, some of those results
are highlighted next. Our numerical results have successfully reproduced the almost linear
voltage vs current behavior seen experimentally by Gamero et al, [52).
Furthermore, our
simulation has shown that if the applied voltage is reduced beyond 20 % of the starting
voltage the cone-jet becomes unstable as seen experimentally. Our results have been used to
18
evaluate the length scales proposed for several authors for the transition region. Our results
show that the best agreement is obtained with Rg, [17]. On a side note, our comparisons
for the transition region with the various length scales, reproduce the trends observed
by Gamero and Hruby for the jet breakup radius scaling, [22].
We present the energy
breakdown for the cone-jet, for various liquids and flow rates. The detailed information
obtained for the cone and jet sections has been used to estimate the accuracy of simplified
submodels for the E-field, and jet radius.
Other results have shown, as expected, that electrode distance is a secondary variable
with weak effect on emitted current.
Similar numerical results have shown that needle
length may under certain circumstances have an effect on cone-jet behavior.
Initial studies for the onset of ion evaporation have shown some of the expected behaviors. Including, increasing ion evaporation current density with diminishing flow rate.
However, our initial results have not been able to reproduce the ion current vs flow rate
behavior seen experimentally.
1.4
Organization
This thesis starts in chapter 2 with a brief summary of the history of colloid thruster
development and of numerical simulations of free jets. Chapter 3 is a short sketch of the
basic physics and basic electrospray scaling laws.
Chapter 4 develops and presents the
numerical model for the simulation while chapter 5 deals with the numerical schemes used
to solve the system of equations. Chapters 6 and 7 present the results of our simulations
along with comparisons of our simulations to experimental data. Finally, we address some
final thoughts, recommendations, and ideas for future work in chapter 8.
19
Chapter 2
Colloid thruster technology
background
In this chapter a short introduction to previous research on colloid thrusters is presented.
The chapter is broadly divided in two major areas: the first refers to general history of colloid
thrusters whereas the latter part of the chapter emphasizes the history of the numerical
simulation of free jets, both charged and uncharged. This review is by no means complete,
but it is hoped that it will help to set the context for this thesis. Where appropriate,
references will be given for books, and review papers that treat each subtopic in more
detail.
2.1
2.1.1
History of colloid thruster development
Colloid thruster technology- 1960's
Pioneering work by Zeleny showed that a stable single cone spray could be obtained from a
liquid issuing from a thin tube 65]. The existence of this stable operating mode, however,
occurred only under certain flow and voltage conditions. It was only until 1964, that the cone
structure was theoretically explained by G.I. Taylor [62]. Taylor found that for a perfectly
conducting liquid a conical equipotential surface was formed. The surface was the result of
a balance between electrical and surface tension stresses at its surface. His analytical model
predicted that the cone semi-angle would be 49.31 which he then confirmed experimentally.
Nonetheless, his studies were not able to explain the formation and the physics of the thin
20
Figure 2-1: Colloid thruster operating in the single cone-jet mode. Notice the cone transition
section, breakup and plume, Courtesy P. Lozano, [47]
, I/
ka
Figure 2-2: Three successive frames (in intervals of 1/64 s) showing a drop of water (which
appears bright) forming a Taylor cone surrounded by oil. The first frame shows a comparison
with the theoretical prediction of the cone angle; the other frames show the drop in two
phases of its oscillation around this cone (Taylor, [62]).
21
Figure 2-3: "Soap film, microsecond exposures of successive stages (a) of jet formation,
(b),(c),(d) subsequent collapse." Reproduced from G. I. Taylor, [62]
Figure 2-4: Taylor cones under various operating conditions. Liquid
conductivity K = 0.07 Si/m. Courtesy Mangoubi and Lozano [50]
22
Ethylene-Glycol,
A
B
Figure 2-5: A) single cone-jet regime. B) Highly stressed regime (seen at higher voltages
than the single cone-jet regime)
jet which emerged from the cone apex. Numerous semi-empirical models tried to explain
the jet physics but the results were often contradictory.
Contemporary to Taylor other groups worked on developing colloid thrusters for propulsion purposes. Krohn and coworkers started experimenting with Liquid Metal Ion Sources
(LMIS) and organic liquids in a variety of configurations aiming to identify suitable configurations of electrodes and liquids for propulsion purposes [40]. Work by Cohen , Shelton
and others at TRW incorporated for the first time Time-of-Flight (TOF) techniques to
the analysis of colloidal thruster plumes [39, 8]. At this time much of their work focused
in achieving the highly stressed regime (multiple cone-jets per emitter) as opposed to the
more amenable single-cone-jet regime (see Figures 2-1, 2-3, and 2-4, 2-5). At the time, col-
loid thrusters were envisioned for main on-board propulsion purposes requiring a relatively
high thrust density and therefore high operating voltages (order of KV). Their inability to
compete with Kaufmann ion engines, which had the required performance at lower voltages,
and the general downturn of the space program after the Apollo era contributed to their
disappearance. Research in colloid thrusters after the 1970's is almost nonexistent.
2.1.2
Colloid thruster- 1990's- present day
In 1989 the discovery by Fenn et al that large organic molecular ions could be extracted
intact from an electrostatic electrospray revolutionized mass spectrometry [12]. Up to that
point mass spectrometry had been limited to materials which could be introduced as a vapor
in a gas without decomposing in the ionization process. The renewed interest in electrospray
technology spurred research in the area and rekindled its propulsion application.
23
In 1994, basic scalings for the pure droplet regime, were published by Fernandez de la
Mora and Loscertales. Their work gave simple non-dimensional scalings for current vs flow
rate and liquid properties [13]. Alternative scalings for the current (I) and droplet size were
obtained by Gafian-Calvo (1997), [15, 17]. The results from Gaiin-Calvo's model suggested
that the polarity of the liquid (i.e. relative permittivity) played no role in determining the
current and droplet size of the electrospray.
Many studies of the cone, transition, and
jet region have been carried out by several authors such as (Gaiian-Calvo[16], Cherney[6],
Higuera[28]) resulting in estimates of the variables as function of liquid properties and
operational parameters. However, the complexity of the phenomenon is far from completely
understood, and remains an active area of research.
Research in the 60's showed that electrosprays could emit both electrically charged
drops and solvated ions, [8]. Three main regimes have been observed: pure droplet, mixed
ion-droplet and pure ion (the latter only in the last three years). The appearance of each
depends on the liquids employed and operational parameters such as flow rate and applied
voltage. Experiments with a few organic liquids such as formamide (K ~lSi/m) operating
at low flow rates q :: 1 showed the existence of ions evaporating from the liquid surface.
Work by Gamero and Lozano has shown that field-enhanced ion evaporation in organic
liquids behaves in similar manner as that of liquid metal ion sources (LMIS), [21, 49].
Further information on the history of colloidal thruster development can be found in
the thesis of Jose M. L6pez-Urdiales, [45], a review paper by Martinez-Sanchez et al, [52],
and the review by Grace and Marijnissen, [24]. Detailed information on mass spectrometry
can be found in the book by Pr6kay, [58]. A comparison between organic electrosprays and
LMIS is provided by Forbes, [14]
2.2
History of numerical studies of free jets
Computational fluid dynamic simulations have progressed significantly in the last few years.
In many cases these simulations compare quite accurately with experimental data and in
certain areas have actually displaced experimental work altogether. The numerical simulation of free surface flows, on the other hand, remains as a challenge and active research
area. A brief introduction to the topic of free jets is given next.
24
2.2.1
Non-charged free jets
In this work we are particularly interested in free jet flows.
These flows have various
properties which make them difficult to model numerically including: variable flow geometry
(in fact it is part of the solution and is prone to instabilities), breakup events (for free jets)
as well as important topological changes [37, 36]. During the breakup event of a nanojet
it is not even clear if near the breakup point continuum fluid mechanics gives a suitable
description of the governing flow physics. Other studies have shown that these flows are
prone to develop cusp singularities.
It appears that surface tension effects present little
resistance to the formation of singularities. Instead, it seems as if the coupling of the bulk
fluid dynamics and the surface evolution make the whole simulation much more sensitive
to noise, and numerical instabilities of which little is known.
Various approaches have been used to numerically model free jets, and here we will
only present a brief description of the main types, as discussed by Eggers, [9]. Three main
techniques have been used in the past to model free jets : boundary integral formulations,
Navier-Stokes formulations, and one dimensional approximations (inviscid and viscous).
Boundary Integral Formulations
This formulation only evolves the surface of the fluid and is limited by the fact that it can
only be applied in those cases in which the governing equations are linear (i.e. its Green's
function is known), [9].
Fluid equations have linear properties for two limiting cases :
inviscid irrotational flow and highly viscous flow. A nice property of this formulation is
that all the information is carried by the surface and there is no need to calculate the
interplay of the internal flow and surface evolution. An obvious drawback of this approach
is that it will either neglect inertial or viscous forces both of which may be important.
In the case of inviscid irrotational flow, a high frequency nonlinear numerical instability
(sawtooth) appears. The origin of this instability has not been identified, a study by Moore
1983 suggests that the equations themselves may not have a regular solution (independent
of the numerical scheme employed), [54]. Numerical procedures such as filtering, numerical
viscosity and regridding have been used to counter this instability with varying degrees of
success.
In the Stokes approximation, the fluid problem is much less sensitive to perturbations
25
Computation
Figure 2-6: Breakup of a filament into satellite and subsatellite drops. As the viscosity ratio
1 is decreased, the final state of the system has an increasingly complicated fractal structure
(Tjahjadi, Stone, and Ottino, [63]). The pictures are, from top to bottom, for 1 = 0.01,
0.067, 0.4, 1, and 2.8.
and no numerical instabilities have been reported. Stone and Leal used this approach to
study drop breakup with excellent results as shown in figure 2-6, [61, 63].
Navier-Stokes formulations
In contrast to boundary integral models, the Navier-Stokes formulations track the surface
evolution as well as the internal flow dynamics. The evolution of both is very complex and
complicated by the fact that they are coupled. Practical concerns also arise with the movable
surface which implies deformations of the computational domain. At pinch points the full
Navier-Stokes equations need to be solved since at these sites both inertial and viscous effects
are important. In general the major problem with Navier-Stokes formulations is that they
are too complicated if analytical solutions are to be obtained. For this reason appropriate
simplifications to the Navier-Stokes equations are sought. The goal of these approximations
is to obtain a simplified set of equations which capture the essential nonlinear physics. One
situation that is amenable to such an approximation is the case in which the liquid jet
becomes long and thin. These approximations are treated next.
26
Figure 2-7: Comparison of experimental and numerical solutions with the lubrication equations of Eggers and Dupont, [10]. The lines represent the profile at various time distances
and at the time of pinchoff.
Quasi-one dimensional approximations of N-S
These approximations were originally inspired by shallow water wave studies by Peregrine, [56].
Lee studied the inviscid nonlinear liquid jet dynamics which presented spurious singularities
which in turn may have affected the pinching process, and contaminated the results, [42].
In 1976 Green studied the viscous form, using for this the so called Cosserat equations.
These equations can be obtained by appropriate averaging over slices of the fluid, [25]. A
systematic derivation of these equations from the full Navier-Stokes equations was first
presented by Dupont in 1993, [10]. Next we reproduce their basic formulation, since our
numerical model will be based on a similar approach.
Eggers and Dupont started from
the axisymmetric Navier-Stokes equations in cylindrical coordinates. Considering that the
study will focus on thin columns of liquid the velocities in the radial and axial direction are
represented with a Taylor series expansion with respect to the radius, r and complemented
by a radial velocity which satisfies conservation of mass:
27
v |z (z,r)
=
v
=
,1
(z, r)
vo + v2r2 +. ..
1,
Vo-
(2.1)
12
2r
-...
(2.2)
(2.3)
Note that the primes denote derivatives with respect to axial direction z. The pressure
is expressed in the same fashion:
p(z,r) = po + p 2r 2
(2.4)
±...
Using these expansions the Navier-Stokes equations are solved to the lowest order in r.
The resulting simplified equations are:
Oh
-
at
_
=
h
l Ov
-vA-- hOz
2 Oz
(
1
(2.5)
hz
P ,Y{h
=
(1+ h2)1/2 ~(1+ h2) 3 /2}(2.6)
av
at
=
3V2 2vz)z
- avz9-z ap + T(h
O z
(2.7)
where r = h at the liquid surface, and the subscript 0 has been dropped. The boundary
conditions for this system are given by:
h(L±,t)=hk
(2.8)
v(Lt,t) =v±
t(2.9)
The properties of these quasi-one-dimensional equations are that near the breakup
(asymptotic limit) they approximate the correct behavior and have also been used successfully away from it by Brenner et al, [1]. The total energy of the system only decreases and
in the case of equilibrium surfaces they reproduce the behavior of the full Navier-Stokes
equations, corresponding to the minimum potential energy. A secondary benefit comes from
the stability arena, in which it is shown that short wavelength perturbations will not grow
28
and the dispersion behavior obtained reproduces that of the full Navier-Stokes, [1]. The
results obtained with this model gave results that compared very well to experiments, see
figure 2-7.
A thorough review of uncharged free jets is given by Eggers (an important part of this
section is based on that work) including history, experimental, analytical and numerical
work, [9].
2.2.2
Charged free jet simulations
Numerical modeling of electrospray phenomena has been an active area of research for many
years. Many models have been developed, some dedicated to LMIS, some for liquid menisci,
and some others meant to study the cone-jet transition and its the downstream plume.
In recent years several numerically based models have been presented to simulate the
cone-jet transition region. The numerical work in this area may be traced back to the model
of Eggers and Dupont [10], and Brenner et al [1], for gravity and pressure driven fluid jets.
The success of these and subsequent models suggested that a similar approach could be used
for electrified jets. On electrified jets several models have been presented such as the static
model of Pantano et al [55], the hybrid model of Gaign-Calvo [16], and the asymptotic
model for the low flow rate limit of [6]. As of this writing several numerical models have
been published such as the works of Hohman et al [32], Hartman et al [27], Yan [64], and
Feng [11].
In 1994, Pantano and coworkers developed a zero flow limit simulation of the electrospray problem [55]. Pantano constructed an axisymmetric model which included: the needle
and extractor as well as a perfectly conducting liquid menisci.
In this case the goal was
to predict the shape of the menisci for different voltage settings. The results obtained, he
reasoned, could be used as a leading order term in a perturbation scheme of the electrospray
problem. A later study by Gafiin-Calvo (1998) employed a hybrid experimental-numerical
technique to calculate the surface charge of jets as well as other parameters, [16]. He experimentally obtained a digital representation of the jet section which was then fitted using a
hyperbolic regression (8th order) thus obtaining the jet shape in close analytical form. The
jet shape was then fed to a quasi-one dimensional flow model from which the surface charge
characteristics were calculated. This information along with measured current for each case
allowed him to develop scaling relations for the emitted current (Eq. 3.47), droplet diameter
29
Figure 2-8: Strobe photograph of the overall picture of an unstable jet. The shutter speed
is 3 ps. On large scales, it appears as a whipping rope, with blob-like varicose protrusions.
From Hohman et al FIG. 11., [32]
(Eq. 3.26) and surface charge density.
In 1999, Hartman et al developed the first simulation of a full cone-jet, [27].The model
used by Hartman assumed a plug flow profile for the liquid. Hartman's work obtained
results for Ethylene glycol, n-Butanol, methanol and heptane. The highest conductivities
were in the order of K = 0.2 mSi/m.
Hohman et al developed a simulation to predict the behavior of electrospun polymer
fibers, [32]. These fibers are made by forcing a polymeric fluid through a needle under the
action of an applied electric field. The electrospinning phenomenon is similar to that of
electrospraying, and Hohman's work built on previous electrospray studies. In this case
Hohman's model is basically an extension of the model of Brenner et al for uncharged jets.
Hohman et al compared their results to experimental data and noticed that the electrode
setup had an important effect on their results. Significant improvement of their results were
obtained when the effect of fringe fields at the liquid-needle interface was added to their
simplified electrostatic model. This work concluded that the whipping jet instability (see
figure 2-8) results from an interaction of the surface charge of the jet and the externally
applied E-field. This whipping instability suppresses the varicose instability (of typical free
30
jets) allowing the formation of long thin fibers.
The third work to appear in this area is that of Yan et al. This work followed on the
steps of that of Hartman et al. The major difference is that Yan introduced a parabolic
velocity profile in his formulation as opposed to a plug flow profile used by Hartman. This
group's work reproduced the experimental data of Hartman et al, [27].
In 2001 Feng presented a new simulation for electrospun fibers, [11]. This work only concentrated on the initial thinning of the fiber due to the electric stretching without actually
simulating the whipping instability. Feng's work built on the previous efforts of Hohman
et al contributing a more accurate representation for the electrostatic fields, and a careful
comparison for non-Newtonian and Newtonian jet behavior. The important contributions of
Feng from the electrospray perspective are: A) was able to explain and clarify the ballooning
instability seen by Hohman's electrostatic model and B) performed a series of parametric
studies of the surface charge density boundary condition. For this condition he concluded
that it was only relevant within a small region (boundary layer like behavior) still far from
the cone-jet transition zone,(see section 4.6).
One of the more recent papers on the numerical simulation of colloid thrusters is that
of Higuera, [28]. In this work Higuera simulates the cone-jet transition section of the free
jet. He uses asymptotic approximations both upstream (instead of actually simulating the
needle-exit condition) and downstream (no jet breakup, infinitely thinning condition) of the
transition as his boundary conditions. His results reproduce the basic current scalings of
Fernandez de la Mora and Loscertales as well as those of Ganan-Calvo,
[13, 15]. Higuera
includes a very rich discussion on the detailed scaling of various flow variables: surface
charge, jet radius, E-fields, etc.
A good review paper on the leaky dielectric paper used for describing charged free jets
is that by Saville, [59].
2.2.3
Mixed ion-drop regime simulations
Original research on ion emission came about from studies of the physics of charged drop
generation and evaporation. Research on ion emission from liquids dates back to the mobility measurements of Chapman, [3]. In his experiments he generated (by spraying and
bubbling) charged droplets of aqueous solutions with various concentrations of dissolved
salts. His results led him to suggest that the electrically charged drops were singly charged,
31
[3]. Further work by Iribarne and Thompson [33] confirmed that the high mobility carriers
seen evaporating from water droplets were singly charged clusters. Their experiments also
showed that depending on the drop size ions separated by evaporation or by Rayleigh's
instability (Coulombic explosion for drops). Iribarne and Thompson proposed a theoretical
model for the field enhanced ion evaporation based on absolute reaction theory and included
estimates of the solvation energy of the extracted clusters. This model theorized that the
potential attracting the escaping ion was due to polarization of the droplet by the ion.
The group of Fenn and coworkers [12] realized that using an electrospray it was possible
to generate electrically charged droplets of complex organic molecules. These drops were
then passed through a drying chamber which further reduced their size increasing its electrical field which led to the desorption of solvated ions or Coulombic explosions. The resulting
quasi-particles would then enter a mass spectrometer for analysis. This novel technique
allows the analysis of proteins and other delicate polar molecules without destroying them,
effectively revolutionizing the mass spectrometry field.
Work by Loscertales and Fernandez de la Mora [46], confirmed that the mechanism for
ion extraction is field evaporation. They proposed a new extraction model based on image
potential theory (effectively a Schottky type model) and compared it to experimental data
with good success. Loscertales and Fernindez de la Mora also obtained estimates for the
drop field intensities required for field evaporation which range between (E ~ 1 - 2V/nm.)
Further work on electrosprays with high conductivity (K ~ 1Si/m) and low flow rates
showed that an important fraction of the collected current came from ions. Detailed experimental research in this topic by Gamero-Castanio, Fernindez de la Mora, Lozano and
Martinez-SAnchez showed that solvated ions could be emitted from the cone-jet transition
region, the jet breakup region and the inflight droplets, [20, 21, 48, 49, 47]
Very little numerical work has been done on the mixed ion-droplet regime, and onset of
ion evaporation. The subject is much more developed for liquid metal ion sources (LMIS)
where the currently is only ionic (plus occasional droplets). For a more detailed discussion on
the LMIS subject see [57]. For organic liquids, the work of Higuera is one of the most recent,
[29]. In this work Higuera extends the basic cone-jet simulation of his previous electrospray
paper ([28]) to the onset of ion evaporation. Higuera based his ion evaporation model on
a Schottky type model (thermionic emission), but due to the numerical difficulty of the
problem had to artificially reduce some of the parameters (see ch. 7). The results obtained
32
with these artificially low parameters, however, reproduced qualitatively the behaviors seen
experimentally by Fernandez de la Mora. The theory of ion desorption is treated in depth
in the paper of Gomer, [23] and the previously referred to Pr6kay book, [58]
2.3
Summary
In this chapter a brief overview of the history of research on colloid thrusters has been
provided with emphasis on the theoretical modeling work. Starting from the pioneering
work of Zeleny at the turn of the century to the most recent developments in the field. In
the next chapter, the basic physics of electrosprays will be presented.
33
Chapter 3
Physics
3.1
Taylor cone
It is well known that when a liquid drop is subjected to a sufficiently high electric potential
its surface experiences an electrostatic pull which if strong enough will deform the surface
of the liquid. The surface deformation, is triggered by small perturbations of the surface
(ambient vibration), which in turn enhance the local electric field (through charge concentration), further increasing the electrostatic pull in a cascading effect, [51]. If the applied
potential is high enough the surface deformation will start to resemble a liquid cone (see
figure 3-1). A limiting case is reached when the apex of the cone approaches a point, i.e. a
singularity. It is at this point that a high speed thin jet is observed to emanate from the
apex of the cone, see figure 3-2.
The first explanation of this phenomenon was offered by G. I. Taylor, who in 1964
Obtained the first analytical model for the cone structure. He constructed a model of a
perfectly conducting liquid cone at equilibrium (with no jet from the apex) for a given
Figure 3-1: Ethylene Glycol Liquid menisci. Left meniscus V=O V, right meniscus V =
2.5KV with superposed spherical coordinate system, photos courtesy Paulo Lozano, MIT
34
Figure 3-2: Structure of the cone jet for Ethylene-Glycol, r7
Fernindez de la Mora and Loscertales [13]
=
2, Re=10.2, courtesy of
ct
V
Figure 3-3: Conical coordinate system
applied potential, Vt. Taylor hypothesized, that the conical shape resulted from a perfect
equilibrium of two opposing forces: the liquid's surface tension and the electrostatic stress
on the liquid's surface. The surface tension force per unit area can be described as
fat = 7-Y
(3.1)
- 1
W1 +R2
where -y is the surface tension coefficient for a given liquid and R 1 and R 2 are the
principal surface radii of curvature.
For a cone the curvature of the normal section is
R, = cot O/r while the second curvature tends to zero (R 2 --
35
o) leading to
cot 0
r )(3.2)
fe =
The electrostatic stress for a liquid is described by the classical equilibrium model put
forth by Landau and Lifshitz [41]. We apply this thermodynamic model to our problem
even though it is clearly out of equilibrium. A better option would be based on liquid kinetic
theory, however, it is poorly developed and therefore the thermodynamic based model is
preferred. The electrostatic stress tensor is given by :
TE = ECoEE
-
2
co 1 -
(E-
1
6\(p)T (-
J)
(3.3)
-
where e is the liquid relative permittivity, co is the free space permittivity, E is the
electrical field, and J is Kronecker's delta. For the normal and tangential directions the
electrical stress components are obtained by subtracting the outer and inner stresses and
neglecting the (dE/dp) contribution leading to:
=
t-Efil
11n.
If
TE
ll
=
[(E.)
-
e(E.) 2 + (E - 1)Et]
eoEt(En - cE) = -Et
(3.4)
(3.5)
where Es is the normal external electric field, Et is the tangential field and En is the
normal internal field. The last two are zero for a perfectly conducting liquid, simplifying
the expression to:
(E )2
fne=lnf-TEn| =
E. ill= 0(37
=|n - TEt3.7
fte fie~iZ
(3.6)
Equating the normal electrostatic stress to the surface tension force per unit area leads
to
1 cot a
=
IoE2
(3.8)
Inherent in this force balance is the assumption that the internal pressure of the liquid
36
is zero.
For a perfectly conducting material (in this case liquid) the surface is an equipotential,
and the field is described by Laplace's equation, V 20
=
0. Laplace's equation for the conical
section (with symmetry about Wp) is:
V
=1
a
r2Or
200)
or
+r2
1
a sin OL)
0
sin 0 70
(3.9)
with 0 measured from the inside the conical section. This problem has known solutions
in terms of Legendre polynomials given by :
0 = AP,(cOS)r"
(3.10)
# = AQ,(cOS)r"
(3.11)
Another possible solution is 1/r'+1 instead of r' but it diverges too fast as r -+ 0. The
Legendre polynomials present singularities at 0 = 180* for the P, and at 9 = 00 for
this case we choose the
Q,
Q,
. In
family of solutions since the singularity is then inside the cone.
The parameter v still needs to be determined, this is done by calculating the normal field
from the potential solution
En
r 09
A
Q
d(cos0)
sin0
1(312)
ri-v
The key here is to notice that in order for the normal E-field to equilibrate the surface
tension effect the exponent has to be v = 1/2. The complete solution is then:
0 = AQ 1 / 2 (cos )r1/ 2
(3.13)
The solution shows that the potential 0 will vary with r. In order to satisfy the equipotential condition at the liquid surface the potential has to be independent of r. This condition is met if
Q1/2(cos9)
= 0, in other words the root of the Legendre function.
The
condition is met for 9 = 49.29* thus predicting the semiangle of the perfectly conducting
cone. Notice that this angle is independent of the applied potential, and liquid properties.
Taylor's prediction has been experimentally corroborated,[62]. The constant A is solved for
37
Extractor
LtL
Needle
V
Figure 3-4: Meniscus and capillary arrangement. L is distance between needle and collector,
Lt is the needle length, and ho is the inner radius of the conducting needle.
by substituting expression 3.12 into 3.8 for 6
A=
=
2y cot aT
CO
aT = 49.29*
1
Q'1/2 (cos aT)
(3.14)
The Taylor cone and its angle (aT) are however not an exact representation of what
happens with any liquid when nonzero current is flowing, since liquids have finite conductivities, and near enough the tip the potential cannot remain constant. Another issue to
consider is that under Taylor's model the potential difference is set between two equipotentials, that of the liquid cone and that of the external electrode, the shape of which is that
of one of the iso-lines of r 1 / 2Q 1/2(cos
6). In experiments, the actual electrode setup may
not replicate the geometry of Taylor's model. Another important effect is the absence of
the charged jet in Taylor's model. The charged jet will modify the potential distribution
of the conical section leading to deviations from Taylor's angle. Given these observations,
it is conceded that the Taylor model was and is a very important step in understanding
the behavior of liquid cone-jets, and is in fact quantitatively correct in a wide intermediate
range of r, between the vicinity of the jet and some distance from the cone's base.
3.2
Starting potential
In the preceding section reference was made to the critical applied potential. This potential
if reached would result in the Taylor cone (actually it overshoots the Taylor cone by about
38
10 %, since it takes more field to overcome a spherical than conical meniscus), and in
practice it is an approximation to the minimum potential needed to develop a jet from the
cone's apex. From a practical standpoint it is useful to make an estimate of this critical
The starting voltage is calculated based on an estimate of the normal electric
potential.
field of an equipotential surface with a round tip (described by a confocal hyperboloid in
prolate spheroidal coordinates) with respect to a ground plane (symmetry plane of said
coordinate system). For a liquid meniscus formed at the exit of a conducting needle with
inner radius ho we can assume that the radius of curvature Rc will be of the same order as
ho:
V=
i2n(4L)
(3.15)
where -y is the liquid's surface tension coefficient, and L is the distance between the
exit of the needle and the ground collector, see figure 3-4. This formula only gives an
approximation to the required starting voltage(experimentally verified within 10%) for each
simulation but thus providing a reasonable starting point for the simulation, [51].
3.3
Cone-jet physics
Once the emission condition for the electrostatically accelerated jet is reached, a number of
new parameters arise. The most important ones are flow rate and collected current. Significant previous research has been done to better understand the physics of jet formation,
jet radius, emitted current, etc., and how they relate to liquid properties and operating
parameters. In what follows we include a summary of the basic scalings for these jets.
3.3.1
Cone-jet transition length scales
The basic physics of colloidal thrusters has been addressed in numerous experimental and
theoretical papers. These papers have successfully yielded basic scaling laws for the cone
and jet regions as seen in the works of Fernandez de la Mora & Loscertales [13], GafinCalvo [15, 19, 16, 18, 17], Hohman et al[32], Cherney [6], Higuera [28], and Clopeau et al
[7] to name a few.
The basic model and scalings of Fernandez de la Mora & Loscertales [13] will be quickly
reviewed as background for later discussion. Two basic assumptions are at its root: (a) The
39
static Taylor equilibrium is maintained to the point where the flow time becomes shorter
than the electrical relaxation time; beyond this, the cone transitions quickly to a jet, (b) At
the same point, the current, which in the cone was carried by the bulk conduction (conduction current, Ica) transitions rapidly to surface free charge convection by the accelerating
liquid stream, (convection current, Ico). Assumption (a) yields a characteristic distance r*
from the apex where flow time is of the order of the electrical relaxation time:
r *3
CEO
-- = -K
Q
where
Q is
(3.16)
the flow rate, and K the liquid conductivity. Inertia has been conspicuously
absent from this discussion. Yet, a non-dimensional factor 72 = inertia/capillarityat the
neck was found empirically to play an important role, in that the minimum flow rate for
stable cone-jet structure was found to be given roughly by q = 1. In more detail,
2=
77-yr
(3.17)
and for r = r*, using 3.16,
772
2pKQ
(3.18)
pKQ
The final aspect of the Fernaindez de la Mora model that we want to mention here is that
it predicts a jet initial radius of the order of r*, however, Fernandez de la Mora concedes
that under 7> 1 conditions inertia may play a role in the transition. For this reason an
alternate length scale is defined. As the liquid flows towards the cone apex with a local
radius h and average velocity u, the dynamic pressure pu2 ;:pQ2 /h 4 at some point becomes
comparable to the capillary pressure
-
y/h. This cross-over occurs at a characteristic radius
Rd=
(PQ2)/
3
(3.19)
and then pu 2 > y/h at smaller radii. In reality because Bernoulli's equation applies,
from zero pressure far upstream,
p +1pu2
2
40
~:0
(3.20)
so the significance of the "dynamic pressure" is a liquid bulk tension (negative pressure).
At h < Rd, this bulk tension replaces capillarity as the force resisting the electrostatic
surface traction. However, the liquid dynamics under these conditions may be unstable
with respect to shape oscillations.
A theory of this instability is lacking, but a simple
argument will make it plausible: if the flow is temporarily restricted (by pinching of the
cone's surface, for instance) the velocity will decrease and by Bernoulli's equation, the
pressure will increase (become less negative). At the same time, the electrostatic surface
traction may actually increase, if the deformation responsible for the pressure increase also
creates convexity. The combined effect is to accelerate the fluid outwards strongly, both
dominant forces cooperating.
This (admittedly sketchy) argument leads us to postulate that, under conditions of q > 1
the jet may actually form at h ~ Rd rather than at r = r*. The direct numerical evidence
for this distinction is not clear enough, since our results cluster close to 77 = 1, where both
distances are equal since they relate by
92/3
-Rd
(3.21)
r*
A third length scale Rg has been proposed by Gafian-Calvo [15, 17]. This length scale
can be derived from a simplified set of electrospray equations (valid near the transition
region), as given by Gaiian-Calvo [16]:
eoE"
(3.22)
pQ2__
EoEnEt
(3.23)
R$o0Ro
oQEn
R2KEt
(3.24)
(
Et
)1/2(3.25)
EoLo
Eq. 3.22 establishes that the pressure depression (related to dynamic pressure) at the
transition is balanced by the electrostatic pull. Eq. 3.23 approximates the axial momentum
equation where the dominant terms are the nonlinear convective term and the electrostatic
acceleration term. Near the transition the convection and conduction current become equal,
41
Eq. 3.24 . Finally, Eq. 3.25 approximates the tangential electric field near the transition
arguing that its behavior is dominated by the cone contribution.
These four equations
comprise a closed system for which we can solve for R,=- Rg resulting in:
1/2=_ 72/3
1(=1/6YK}
Rg R9=-3
-
2
r*Rd 1/
k.E1/3)(.6
which can also be expressed in terms of r* and Rd. In subsection 6.2.4 we will address
again the transition length scale issue and see how r*, Rd, and Rg compare to numerically
obtained results.
The scaling analysis by GaiiAn-Calvo also obtains two non-dimensional parameters (in
his notation, [17]):
=
a
=-
(3.27)
pKQ =E2
'YEo
- E7(3.28)
23
(3.29)
a, and a, which have the same function as 7.
In other words both a's define non-
dimensional flow rates (Q/Qsce). The difference resides in the reference (or scaling) flow
rate, Qacale. For a, the reference is derived from equations (3.22-3.25) and is given by:
Qscale =
-.
(3.30)
The second group a ,, describes a different flow behavior than that given by equations
(3.22-3.25). In this case equations 3.22 and 3.23 are substituted by:
~"
A
~
,P
R3Lo
(3.31)
EoEnEt
Ro
(3.32)
respectively. This means that for highly viscous fluids (where viscous effects dominate
over inertia) the viscous stress will dominate over the electrostatic pull near the transition
region. Solving for
Q in this
new set of equations results in:
42
(3.33)
Qscale =
which in turn defines a. = Q/Qcale of equation 3.28.
The range over which these
non-dimensional flow rate scalings are valid is discussed in detail in section 3.3.3
3.3.2
Viscous effects
Previous studies have addressed the importance of viscous effects on cone-jet electrohydrodynamics such as Fernmndez de la Mora and Loscertales [13], Gafian-Calvo et al [18, 17],
and Cherney [6].
Fernaindez de la Mora and Loscertales define the Reynolds number in
terms of r*:
Re* =
= ll04/3
vr*
(3.34)
where v is the kinematic viscosity, and the nondimensional group 1, is:
/ 2
A
KI
1/3
(3.35)
3
which only depends on the liquid properties and serves as an approximate measure of
the ratio of inertial to viscous forces (since 7 ~00(1)) for a given liquid.
Scaling arguments by Fernandez de la Mora and Loscertales argue that viscous effects
become negligible when the non-dimensional viscous parameter Zf(e)/e is small:
z =H
Z
f
77- 2/3
< 1
=Re
772
(3.36)
(3.37)
where Z(equation 17c, [13]), is a measure of the radial variations in the axial velocity
and f(e) is an experimentally obtained parameter of order 20,(equation 18a, [13]). Viscous
effects tend to make the velocity profile flat while electrical traction shows the opposite
effect.
Gafin-Calvo defines a Reynolds number based on his two non-dimensional groups
(Eq. 3.27, 3.28):
43
/
Res
=
\4/3
a
.(3.38)
a,
which can be rewritten in its classical form:
Res
= (pK2
L-=
where the length scale is R, evaluated with
1/3
(
-(3.39)
Q = -yeo/(pK)
from 3.22- 3.25.
The importance of viscosity is most readily seen by using the scaling parameters of
Gaiiain-Calvo where viscosity dominates over inertia (Re. < 1) for a, < aA/ 4. Colloid
thrusters and typical electrosprays (such as the results of Fernandez de la Mora and Loscertales [13]) fall in the category of a>, > at/ 4 relegating viscous terms to a secondary role
with little effect on the current scaling, [17]. Viscous effects do become important in other
circumstances such as in the electrospinning of fibers.
3.3.3
Current scaling
(b) At the same point, the current, which in the cone was carried by the bulk conduction (conduction current, IM) transitions rapidly to surface free charge convection by the
accelerating liquid stream, (convection current, I,)
The second assumption of the Fernandez de la Mora and Loscertales model (see section 3.3.1) which describes the transition from a dominating bulk conduction current to a
convected current near the transition region. It is this assumption which allows current
to be calculated.
At about the crossover from conduction (I ) to convection (I ), the
convective current (which rapidly becomes all of the current) is approximated by
I,~
2rhua
(3.40)
where - is the free surface charge density. In order of magnitude, h ~ r*, since both
transitions are assumed to be co-located. Also u ~ Q/irh2 and since surface charge equilibrium has been maintained right up to r*, u- ~ eoE*, where E* is as in Taylor's static
equilibrium (surface tension balancing electric traction)
L=2cot a eor*
44
(3.41)
combining and ignoring numerical factors,
Qv~
M2
(3.42)
and using equation 3.16:
I
-KQ
(3.43)
The factor of proportionality, f was argued to depend at most on the dielectric constant
6, though the details of the finite- width transitions were ignored. In all,
I = fW
where
KQ
(3.44)
f(e) ~ 18 - 20 for e > 40, decreasing more or less linearly to zero for smaller C.
Because of the central role of 77 in selecting the flow rate (we normally want flows close to
the minimum), it is worth rewriting the current expression 3.44 in terms of 7
f-YFL-7 y(3.45)
which points to a convenient scale for current, namely I(77 = 1)
=
Ii
=
f*
(3.46)
A similar current scaling is proposed by Gaiian-Calvo [15, 17]:
(3.47)
7KQ
IGE =
differing from equation 3.45 in that it has no dependence on the relative permittivity
(E).
Gaian-Calvo, mentions that two main regimes exist for colloid thrusters. The first
one, inertially-electrically dominated flows, (IE in his notation) follows the current scaling
of equation 3.47. The second one is inertia-polarization dominated (IP), [17] and scales
current as:
IcP
=
K 2 Q2
(E - 1)EO)
45
1/2(3.48)
Estimate
symbol
Equation
Reynolds number range
flow rate range
F. de la Mora
IE
Ifdm
3.44
Re*<< 1
77> 1
IGE
3.47
ap>
VE
IP
IGE
3.47
ap<« a1 /
IGP
3.48
AY
1
«1
0124
Table 3.1: Main current scalings, [13, 17]. Note that in the case of the F. de la Mora scale
the current scaling holds even for Re* >> 1, showing the same trends as the IE and VE
current scaling.
The IE regime is the most common one and the data of Fernindez de la Mora and Loscertales belongs to this group,[13] while the IP regime has been observed from L6pez-Herrera
data and according to Gafiin-Calvo for the Formamide and water results of Fernandez de
la Mora, [43, 17].
Gafitan-Calvo discusses two viscosity dominated regimes : VE which
is viscosity and pressure suction dominated (originally proposed by Higuera, [28]) and is
typical of highly viscous liquids and of electrospinning. In this regime the current also scales
as given by equation 3.47 while the jet radius scales as [28, 17]:
RVE=
K(3.49)
The last regime is the viscous and polarization dominated regime (VP) which has not
been observed (no published data has been found to comply with this scaling), [17). Numerical results will be compared to these scalings in chapter 6. The domains of applicability of
the scalings proposed by Fernindez de la Mora and Gain-Calvo are presented on table 3.1
In this work results are reported using the Fernandez de la Mora scaling where the flow
rate (Q) is reported in terms of the dimensionless variable 77 and the current is reported in
terms of the dimensionless group ( :
pKQ
pKQ
C
(3.50)
.9
46
(3.51)
3.3.4
Drop characteristics
Downstream of the transition region the jet will continue thinning out and eventually when
it reaches a radius (Rb) it breaks up into charged droplets.
Experiments have shown
that droplet radius (rd) is approximately rd = 1.89Rb, conforming to the Rayleigh-Taylor
breakup prediction for un-charged jets, [51]. Theoretical upper and lower bounds for the
charge to mass ratio of the primary drops are presented next. The upper bound is based on
the maximum charge that a liquid drop can sustain for a given radius. Under equilibrium
conditions the balance of the electrostatic pull vs surface tension for a spherical drop is:
Enoq
)'
2
2
Aireord
-o(E
(3.52)
2-(3.53)
rd
solving for the charge (q) and dividing the resulting expression by the drops mass (m)
gives:
q]
6(eoy) 1 /2
max
(3.54)
prd
this expression is known as the Rayleigh limit for the charge to mass ratio of a liquid
droplet. The minimum bound for the charge to mass ratio is based on a thermodynamic
energy minimization argument which concludes that the minimum charge to mass ratio for
a droplet is half of the Rayleigh limit q/mlm. = 2 q/mlmin, [51]. For a high conductivity
formamide solution (K - 1 Si/m) the highest charge per unit mass measured has been in
the order of q/m = 10, OOOC/Kg, [47].
Our work will not address the droplet formation process. which include complex interactions among viscous, electrical, inertial, and capillary effects and lead to phenomena such
as satellite droplets, long fiber formation, etc. For more information on this see for instance
the work of L6pez-Herrera and Gafiin-Calvo, [44].
47
900
800
700
Itot
600/
500
-
400 -300200 -100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Q [nl/si
Figure 3-5: Flow rate vs current plot showing the onset of ion evaporation. I pure droplet
current (I oc Q1 / 2 ), Itot = I + Ii, Ii accumulated ion current. Courtesy Paulo Lozano, [47]
ion emission zones
Figure 3-6: Possible electrospray ion emission zones
3.4
Mixed ion -droplet regime
Under certain operating conditions the electric field present at the surface of the electrostatically accelerated jet may be strong enough that ions begin to evaporate from it. In
this section we first develop the models used to describe the field desorption of these ions.
Secondly, we describe the typical characteristics of the extracted ions. Thirdly we address
the effects of space-charge on the jet surface E-fields.
The onset of ion evaporation can be seen in a collected current vs flow rate plot as a
departure from the I
Oc Q1/2 behavior typical of the pure droplet regime. This behavior is
depicted in figure 3-5 which shows that as the flow rate decreases ion current
48
(Ii) becomes
of the order or larger than the drop current (I) resulting in It& > I. In the cone-jet mode a
decrease of flow rate results in a higher surface field. If the surface field reaches a threshold
value before the jet instability condition (1 5 1) is reached, ions will begin to evaporate
from its surface. The condition for ions to be emitted can then be expressed in terms of
Q=l < QE=Ecrit. The maximum fields are expected in charge concentration areas, in this
case the cone-jet transition and the jet breakup point. It is these zones that are the most
likely to evaporate ions, see figure 3-6. A maximum field estimate for the cone-jet transition
region is given by,[21]:
Emax =
a-
(3.55)
2/31/
substituting equation 3.18, Emax is reexpressed as:
(/2\
1/6 K 1 / 3
Emax = (YP
1/3
(3.56)
for high conductivity liquids (K ~ 1 Si/m) the surface field may approach 1 V/nm (the
threshold field). A quick estimate (using Eq. 3.56) shows that for a formamide solution
(1 = 1) the required minimum conductivity for ion evaporation is K = 1.3 Si/m which is
of the same order as seen experimentally,[21, 47].
At this very high field ions begin to evaporate from the liquid's surface with an ion
current density (ji). For the case of the cone-jet transition (meniscus), the current per unit
length A is:
A = 27rhji
(3.57)
and the total accumulated ion current is:
ii =
j/
10cos a
dz-
(3.58)
The total collected current is then Ita = I+ I. The ion current density ji is discussed
in more detail in the next section.
49
free space
(D)
E
Figure 3-7: ion desorption from jet surface, image potential
3.4.1
Ion emission mechanism
Iribarne and Thompson developed a model based on absolute reaction theory to describe
the extraction process with good results, [33]. Further work by Loscertales and Fernandez
de la Mora [46] on a more accurate model based on image potential theory, effectively a
Schottky type model for the ion current density, gives:
.
3i =
where,
o-KBT
exp(
mKBG
(3.59)
o is the surface free charge density, KB is Boltzmann constant, T is the liquid
temperature, hp is Planck's constant, Go is the solvation energy of the ions. Wmax is the
Schottky depression which is function of the normal field E*:
Wmax
=
('3rn)1/2
(3.60)
where q is the charge of the evaporated ions. A full derivation of this model is presented
by Lozano [47]. Notice that if Wmax = 0 the process is then thermally controlled. In this
case for formamide (Go ~ 1.7 eV) under standard conditions (T=300 K) Eq. 3.60 results
in negligible evaporation since the exponential term is very small and overpowers the rest.
Once the applied electrical field reaches a threshold limit things change dramatically. An
order of magnitude estimate of the E-field for evaporation can be obtained by equating
Wmax = Go and solving for the field:
50
I
~\
1
0.8 - 1
.
0.6-
-
0.4-
0.2-
0
2
4
6
8
10
sc
Figure 3-8: Space charge effect C8c(Eq. 3.63) on field fraction (E,/E*), Eq. 3.62. Space
charge is estimated based on formamide K = 2.1 Si/m, I = 400 nA, 0 - 800 V, q/m
350 x 10 3 C/Kg, and E,
1 V/nm. (1) Refers to a jet radius h = 10 nm and (2) to a jet
radius h = 30 nm
E* = 41riEOGO
which for Go = 1--2eV requires E* = 0.6 -2.8V/nm.
that only high conductivity K ;
(3.61)
Experimentally it has been found
1 Si/m liquids are able to evaporate ions, for the reason
discussed in connection with Eq. 3.56, [46, 21, 47].
Experimental studies have determined that in many cases the evaporated ions are solvated, [33, 46, 21, 47].
As an example for a formamide solution doped with Nal the
charge to mass ratio of the Na+ ions is estimated (from experimental data) at q/m =
350,196 C/Kg, (471 . Assuming that the ions are singly charged, the evaporated ion mass
is mi = 275 amu. The ion mass is considerably larger than that of the Na+ (23 amu)
which suggests that ions are solvated. The degree of solvation i.e. the number of formamide
molecules (m = 45 amu) per each Na+ is estimated at n. = 4 - 5 [47].
3.4.2
Space charge effects
One of the effects of ion evaporation from the liquid surface is that the in-flight ions will
modify the surface potential distribution according to Poisson's equation V 20
-
Pe/Co
where p, is the charge density distribution. The ion evaporation model is based on a one-
51
4.5 -
4-
2.521.5
11
---
1.5-
I
0.500
-
1
2
3
4
5
n p
Figure 3-9: Space charge effect as given by equation 3.66. Notice the two limiting cases:
Child-Langmuir space charge limited case e, = 0, and weak space charge case, el -+ oo and
6P -+ 00
dimensional approximation derived by Stern et al [60] and reproduced by Prewett & Mair
[571. This model estimates the reduction of the electric field (Laplace field) due to the effect
of space charge. The ratio of the the Poisson field (Ep) to the Laplace field (EG) is given
by
Ep
+(1+ (SC-1
(3.62)
(1 + 3(9c) 1/2
where the dimensionless space charge strength is (,c is given by
c=
3eEEV 2q/
where
ji is
-q/
(3.63)
the ion current density, qB the liquid potential at the extraction point, and
(C
< 1 (weak space
) For the case where Cc
> 1, the result is
q/m the charge to mass ratio of the extracted ions. In the limit where
charge), equation 3.62 becomes Ep = En(1 -
the Child-Langmuir law for a diode (space-charge limited case):
Jd =
d(3.64)
52
and d is the distance over which space charge effects are important.
In figure 3-8 this simplified model is shown and two sample cases are presented. Based
on experimental data for a formamide solution of high conductivity, [29] with two estimates
of the corresponding jet radius h = 10 nm (point 1 figure 3-8), h = 30 nm (point 2 same
figure). Note that this simple calculation shows that the fields may be modified by as much
as 40% and may therefore significantly control the emission characteristics of the meniscus.
Formula 3.62 is deceiving, in that it gives E, in terms of (C,
Ep.
which itself depends on
A clearer representation may be obtained if we reexpress formula 3.63 in terms of
(C = (Ep*/E )2where:
E* =
E/(3.65)
36icE
2q/m
which can be substituted into equation 3.62 and recast in the form:
0 =
=27/8(3.66)
E
e +[
e2 [e+ ±+3]1/2
where e, = EIE*. Equation 3.66 is plotted in figure 3-9. For the space charge limited
case Ep = 0 and the ion current density is that given by Child-Langmuir, formula 3.64
(e' = v/27/4). For the opposite case weak space charge (e0 = e,), actually e-0
->
oo and
Ep --+ oo. There is still the fact that qB enters in E*, and it should not be the full surface
potential, only the (unknown) potential drop in the space-charge region near the surface.
This however, is harder to remove without a full 3D solution.
3.5
Summary
In this chapter a short summary of the relevant physics of colloid thrusters has been presented. The first part of the chapter concentrated on the Taylor cone as precursor to an
emitting electrospray. Once a threshold voltage is reached the cone apex deforms and a thin
jet emerges. Estimates of the size of the cone-to-jet transition region have been presented.
The electrified jet's current is estimated based on analytical and semi-empirical scalings
introduced by several authors. Finally, for some liquids of sufficiently high conductivity
ion evaporation from their surface is observed. The physics of this evaporation process are
discussed and a simple ion evaporation model is introduced.
53
Chapter 4
Cone-jet numerical model
In this chapter we derive the equations for the electrostatically accelerated free-jet flow.
The derivation process is outlined and the simplifying assumptions employed are justified.
The model includes viscous effects, charge relaxation effects, complete free surface boundary
conditions, and the effects of the needle length on the liquid potential. Ion evaporation from
the jet surface is added to the basic model as an extra module. The ion model includes
a simplified space-charge effect E-field reduction model, as well as modifications to the
conservation of mass and charge equations. Finally, boundary conditions for the system of
equations are introduced.
4.1
Computational domain
Previous numerical work on cone-jet simulations has presented two main approaches to the
upstream boundary condition of the cone flow. The first approach favored by Higuera, Cherney and others is to assume that far upstream from the transition zone the cone approaches,
in the limit, Taylor's angle and becomes truly equipotential, [28, 6]. The reasoning behind
this is that considering the disparity of length scales the cone-jet transition will be a local
behavior without any major influence from the external electrode configuration. This idea is
supported by Fernandez de la Mora, who mentions that if the ratio of inner capillary radius
vs jet radius is ho/hj > 100 then the external electrostatics (i.e. electrode configuration)
will have little effect on the transition behavior, [13]. Experimental photographic evidence
presented by Gafi6n-Calvo et al shows that for a variety of flow rates and operating voltages, far from the transition region, the cone will not deviate much from Taylor's angle,
54
figure 2-2, [19].
On the other hand, the zeroth-order cone simulation of Pantano shows
that equilibrium shapes at higher angles than Taylor's are physically possible, [55]. This
is supported by photographic evidence from Taylor's original paper, ([62], Figure 2-2). It
seems that under many circumstances imposing that upstream from the transition region
the cone approaches Taylor's angle is not a bad assumption, but it may not be applicable
to all conditions, as can be seen in Figure 2-3), [62]. Similar pictures have been obtained
by Mangoubi and Lozano, some of those can be seen in Figure 2-4.
The second approach includes simulating the liquid cone-needle interface as done by
Pantano, Yan, Hartman, Hohmann and this work, [55, 32, 27, 64]. These authors observed
that the numerical results were affected by the electrode configuration. One thing to keep
in mind is that in various cases other numerical codes have had problems, as is expected,
at the liquid-solid interface (see section 4.6.)
4.2
Free jet model
The free jet numerical model for the cone-jet transition is based on the concepts put forth
by Eggers, Dupont and Brenner for non-charged liquid free-jets. Their model was based
in a "one-dimensional long wavelength approximation of the full hydrodynamic equations"
as seen in Eggers & Dupont [10] and Brenner et al [1](see chapter 2). Their simulation
results showed very good agreement with experimental data. The model we will present
next extends some of these ideas to electrostatically accelerated jets.
Our simulation reproduces the basic setup of Fernandez de la Mora & Loscertales with
a conducting capillary needle facing a ground electrode at a distance L, as seen in Figure 41, [13]. The liquid is modelled as it emerges from the needle (i.e. flow inside the needle is
not modelled) until it passes through the ground electrode screen.
Most free jet models (electrically charged or not) assume that the jet is axisymmetric
and describe its form in cylindrical coordinates [1, 27, 32, 11]. In our case we have chosen
spherical coordinates, due to the conical form of the problem in the essential upstream
region, [15, 6]. The liquid surface is described using a local axisymmetric (about p)conical
approximation as seen in Figure 4-2. The free-surface boundary conditions are kept exact
in order to preserve the correct equilibrium shapes for the jet but more importantly, for the
cone section as suggested by Brenner et al [1]. The model uses continuum fluid equations
55
Extractor
liquid jet
Needle
2ho
~
L
Figure 4-1: Simulation setup with conducting needle and ground extractor electrode
flow direction
U(z,t)--
h(z,t)
UO(z,t)
z
Integrated region
Figure 4-2: Local spherical coordinate system used for the derivation of the fluid equations
complete with viscous effects.
Laplace's equation is solved to obtain the potential and
The
electric fields, at the outer surface of the electrodes and of the liquid via a panel method.
formulation assumes that the liquid has a constant conductivity. This last point deserves
(ions)
some extra attention, because it is not immediately obvious how the charge carriers
will redistribute themselves in the fluid, given the strong fields and accelerations present,(see
Appendix A of [2]).
4.2.1
Fluid mass and momentum equations
of the
A parametrically described parabolic velocity profile in the r - direction is assumed,
form
Ur(Ua,UO,,a)=-
+(Ua
56
uo)
sin 0
(4.1)
where u, is the velocity at the jet surface, uo is the velocity at the symmetry axis, 9
is the conical angle and a is the angle of the jet surface to the z-axis as seen in figure 4-2.
This formulation will have problems in those cases when a -> 0 and care is taken to obtain
the physically correct limits in such cases.
A corresponding consistent velocity profile in the transverse-direction (increasing 9) is
obtained by substituting the radial velocity profile, of equation 4.1 into the mass conservation equation (in spherical coordinates) and solving for uo
1 -- cosa (2r
rsin 9 Or \
_(1-cos
[
0)(2 + cose0) (ua,-
o)Th(4.2)
J i
sina 2
[U3
The two velocity profiles are now substituted in the r-component and 9-component of the
momentum equation in spherical coordinates and every term is integrated for 9. The force
terms have been projected in the z-direction (axial) in order to obtain a more physically
meaningful expression. The integration in the 9-direction is carried out from 9 = 00 (the
z-axis) to just underneath the surface of the colloid jet 9 < a at constant r.
(term)r2sin9 cos] dO-
(4.3)
r is substituted by r = h/sin a and derivatives are transformed according to
O(
)
00
=
- cos a-O()
1Z
Or
(4.4)
is order to relate the equations to the axial z-direction. A small angle approximation
is made (a << 1) and only terms up to a are maintained (i.e. a", n > 1 are assumed
small). The small angle approximation is not applied in two cases: the surface tension term
of the normal surface boundary condition and for the calculation of the electrical fields on
the surface of the liquid jet. The surface tension terms are kept exact so that the model
equations reproduce the static equilibrium surfaces of the complete equations, [10].
A useful construct is to define two velocities, the mean velocity u and a "shear velocity",
ut:
U
+
-
2
57
(4.5)
U
(4.6)
U-UO
This change of variables is only meant to simplify and compact the representation of
the equations. The resulting equation of conservation of mass is given by
0(h2 )
at
8(h2 U)
Oz
(47)
where h is the local jet radius, and u is the average velocity as defined by equation 4.5.
The net volumetric flow rate at a section is given by
27rh 2
1 + cos a
=uo
E +
(2+ cos a)
3(1 + cos a)
(un- o)(4.8)
Q
from which the small angle approximation yields
~ irh2 u where
Q
is constant in
the steady state, but can fluctuate in space and time during transients. The momentum
equation in the r-direction becomes after much algebra and simplification:
au
19(2 U2 2 )U'a2(u
( 2 +z-+32
5(h 2
2
U'_1Po
+61)
8vu'
+-z h2
+ gz-
(4-9)
and in the 09- direction
Pa-
au'
Po = -21L-oz
(4.10)
Pa denotes the pressure at the surface of the liquid, PO the liquid pressure at the jet
axis, v is the kinematic viscosity, and M is dynamic viscosity.
4.2.2
Charge Conservation Equations
The electrical charge formulation of the system assumes that the fluid is undergoing a
charge relaxation process with constant conductivity K. The movement of charges in the
axial and radial direction is modelled by an internal (Figure 4-3, bulk control volume C.V.2)
charge conservation equation which assumes that there are no free charges within the liquid,
resulting in
58
Figure 4-3: Section of the liquid jet showing the Control Volume C.V. 1 used to derive 4.12
and C.V. 2 used in the derivation of equation 4.11.
2irhKEi
cos a
+
(912irKEth2 \
- I
=0
Oz 1+ cos a
(4.11)
where Ei and Et are the components of the internal electric field normal and tangent
to the surface. The first term of equation 4.11 is the conduction current from the bulk to
the surface (0 - direction) while the second term is the rate of change of the conduction
current in the axial direction.
The rate of charge accumulation on the surface (Figure 4-3, control volume C.V.1) is
comprised of the surface charge convergence due to convection (left-hand-side) and charge
contributed by the normal current density arriving from the bulk,
(ho) + cosa a (hau) = hKE,
where
j.
= KE.
(4.12)
o is the free charge density at the liquid surface, and u" is the surface velocity(u.
=
u + u'). The free charge at the surface interface gives a jump condition between the internal
(E,) and external field (E,) derived from Gauss law : o = E(En - eE),[5]. The tangential
fields are not affected by the presence of free charges and are therefore continuous at the
interface Et = Et' = Et. In the case of a fully relaxed fluid Ei = 0 which reduces the
surface condition to a = EOEn.
59
4.2.3
Free surface boundary conditions
The description of the free surface of the liquid jet requires normal and tangential stress
balance boundary conditions. The normal condition is given by :
AP = P,, - P = -yr + p||n-,r -fi-
1n-|-T E
(.13
where Pa is the pressure inside the liquid just below the surface, P, is the pressure
outside the liquid equal to zero in vacuum, r. represents the local curvature of the liquid, -r
the viscous stress tensor and TE the quasi-equilibrium electrical stress tensor formulation
of Landau & Lifshitz [41]
TE
where 6 is Kronecker's delta.
1C
[O-
2
1
EE -
(4.14)
\op)TI
For the normal and tangential directions the electrical
stress interface condition become:
|n- TE -n|
[(E)
||n - TE t|=(
E):(4.16)
where it is assumed that de/dp = 0.
e(E.) 2 + (e - 1)E ]
-
(4.15)
One modification is made to the equilibrium
formulation of Landau & Lifshitz [41] where for 11
- TE -$ we substitute 0'relazed = 00 E*
for - which is not, in general, fully relaxed.
The normal viscous stress is expressed in spherical coordinates and a small angle approximation is made leaving only the leading terms. This approximation is made on the
grounds that viscous stresses are assumed to be much smaller than the surface tension and
electric stresses.
The normal surface condition is now expressed as
PO
1
=
-Y
/
h+2),2
h (I+
/h2au'a\
1+
60
(9h)2)3/2
h
o )
20[(Eo)2
-
e(E)
2
+ (C
-
1)E2]
where Pa has been eliminated by using equation 4.10. The tangential boundary condition expresses a balance between the viscous shear stresses and the corresponding tangential
electrical stress.
u=
4.3
4y
-Eth(4.18)
Electrostatic problem
The electrostatics of the needle-liquid jet system are described by Laplace's equation (V240
0). This description neglects any space charge effects due to evaporated ions from the conejet surface. In those cases where ion evaporation becomes important a correction to the
Laplacian field is made, see section 4.4. The relationship between the potential 4 and the
external normal field Es is complex since every charge in the system (on the liquid as well
as on the electrodes) influences each local potential. The problem is further complicated
by the fact that while the system of equations is time marched the shape of the jet and
the total charge at the surface will fluctuate thus modifying the potential and electric field
of the system. Laplace's equation is solved using a panel method described in detail in
section 5.2
4.4
Field enhanced ion evaporation
One of the effects of ion evaporation from the liquid surface is that the in-flight ions will
modify the surface potential distribution according to Poisson's equation V2q =-pEO
where p, is a charge density distribution. The ion evaporation model is based on a onedimensional approximation derived by Stern et al [60] and reproduced by Prewett & Mair
[571. This model estimates the reduction of the electric field (Laplace field) due to the effect
of space charge. The ratio of the Laplace field (ED) to the Poisson field (Ep) is given by
equation 3.62 and the ion current density by 3.59.
The rate of charge accumulation on the surface (control volume C.V.1, Figure 4-3) is
comprised of the surface charge convergence due to convection (left-hand-side) and charge
contributed by the normal current arriving from the bulk, j, = KEi, and that emitted by
61
the extracted ions.
0
0
(h-)+ cos a-(houn) = h(j, - ji).
(4.19)
Mass conservation is also affected by the evaporation of ions from the liquid surface.
The effect is captured by the last term of the R. H. S. of equation:
0(h 2 )
&t
4.5
_
2
=- (h u) _-.
-U mj
--
Oz
q
(4.20)
4.20p
Total current computation
The total current carried at a section of the jet is computed as the sum of convective,
conductive, and free ion parts. The surface convective current Icv due to the convected free
charges at the surface of the jet is:
Icv = 27roh(u + u')
(4.21)
where h is the jet radius, u is the average velocity, u' is the difference between the axial
surface velocity and centerline velocity. The bulk conduction current Iad:
Icd
27rKh2 Et
= (1+cosa)(
2 h t(4.22)
where K is the liquid conductivity, Et is the tangential electric field and a is the angle
of the jet surface relative to the centerline. Finally the accumulated ion current is
i=
In the steady state, I
=
j
2rdz.
(4.23)
Ic + Icd + Ii = const, but I depends on time and space during
the transients.
4.6
Boundary conditions
The mass and momentum equations require upstream boundary conditions. For the mass
conservation equation (Eq. 4.7) the radius is set equal to the needle's inner radius (h(0) =
ho). The average velocity u(0) is specified as u(0) = Q/(7rh2 ). The surface charge density
62
boundary condition -(O) (i.e. liquid-needle interface) depends on the details of the electrode
setup as well as the imposed fields as noted by Feng, [11]. To deal with this issue different
approaches have been proposed.
Pantano used an analytical approximation to take care
of the singularity at the interface of his perfectly conducting liquid, [55]. Hohman et al,
experimented with different values for the charge density in the liquid at this point. They
concluded that the better results were obtained when its value was set to zero or near zero.
Another important point made by Hohman was that it is important to include the fringe
fields since experimentally it has been seen that they contribute in determining both shape
and stability of the cone-jet, [32]. Hartman et al noticed that the cone shape calculation
was quite unstable. They noticed that their error minimization code was able to converge
(to zero error) if an appropriate value for liquid angle at the liquid-needle interface was
chosen,[27]. Feng used an approximate model to calculate the E-field (based on Hohman's)
and experimented with various values for the surface charge density boundary condition.
From his results he noted that the actual boundary condition value only had an effect within
a thin boundary layer (0 < z < zbL) far from the transition. For z > zbl the behavior of
the surface charge density became almost independent of its upstream boundary condition,
[11].
In our case, our simulation includes the needle and liquid jet. The surface charge density
boundary condition is updated based on the evolution of the E-field distribution. We have
assumed that surface charge at the liquid-needle interface is fully relaxed and is therefore
given by u(O)
e-oEO,(1).
63
Chapter 5
Numerical Solvers
In this chapter we address the numerical approach for solving the system of equations
derived in chapter 4. The system of equations can be subdivided into two coupled subsets:
the first one describes the evolution of the jet (fluid mechanics) and its electrical charges,
and the second one its potential distribution (electrostatics). Changes in one of the subsets
reflect in the other. Our approach is to solve them sequentially until a steady state solution
is reached. For each subset we present the numerical solver employed, the discretization
schemes used, and the boundary and initial conditions required.
Finally we address the
interaction between both subsets as well as some practical information regarding gridding,
time-stepping, etc.
5.1
Time dependent jet evolution
The system of equations presented in chapter 4 is split in two subsets which are solved
sequentially. The first subsystem is comprised by the time dependent fluid equations, charge
relaxation equations, and surface boundary conditions:
1. mass conservation, Eq. 4.7
9(02 )
_
2t
2. momentum conservation, Eq. 4.9
64
(h2 U)
au
1a(2
+U
-2
+
2
+U'(2u+U'
2 j&Z
)+
Po
p jz
_1
6
(2
8vu'
-2
h -+ 9z
3. surface charge relaxation, Eq. 4.12
(ha) = -cos a
(ha(u,)) + hKE,
4. normal surface boundary condition
2
10
(
=(h1+(1+
-
O [(E.)2
-
h
2au' Ou
23/2
h
19z
E(E') 2 + (e - 1)E,]
5. tangential surface boundary condition
,1r
soEth
4A
5.1.1
Time marching numerical solver
The numerical scheme employed for time marching these equations is a general purpose
fourth order Runge-Kutta. This is a well established time stepping scheme, which employs
an explicit time integration method that has been successfully used in the past to solve
the Euler equations, [35, 30]. For an initial value problem of the form y'
Runge-Kutta scheme is given by:
65
=
f(t, y) the
k0)
=
f(t",y")
k
= f
tn+A
k (3)
=
t + A
-(2)
f
)
Yn+
7
Y+
(5.1)
k2)
k(4) = f (tn + At, yn + k(3)
y(+1=)
+ 2k(2) + 2k 3 ) + k(4))
_-A(k)
_ Y"
6
This is a general version of the scheme. For fluids problems, however, the time dependent
and spatial part can usually be separated as:
d=
-(Y)(5.2)
where the right hand side only has space derivatives and source terms. The scheme may
then be written as:
)=
yn
(5.3)
k (2)=
y" -Atf(())
k)
y
k(4)=
y
yl
y(+1)
At
f
-
At
f(k(3))
-
At(k()
(t )
6
+ 2k(2) + 2k)
+ k(4))
From a stability analysis this scheme is seen to be conditionally stable for central differenced convection equations, [30].
The basic Runge-Kutta scheme described above has
been modified to include residual smoothing, and artificial viscosity terms. The residual
smoothing is in essence an averaging process:
f(y)i = e[f(y)--1 + f(y)i+1] + (1 - 2e)f(y)i
(5.4)
where e is a constant parameter set in this case to 0.1. Hirsch points out that it is
66
sufficient to apply this modification to alternate stages (2nd and 4 th in this case), assuming
that e is large enough, [31]. The numerical or artificial viscosity terms (N.V.T.) are applied
to the momentum's equation right hand side (R.H.S) at every stage of the Runge-Kutta
solver (i.e. RHS(NVT) = RHS + NVT).
The form of the numerical viscosity is that
proposed by Eggers and Dupont [101 and analogous to:
-1
where U = vnvAz where v
=
NVT
(5.5)
> 0. The finite difference version is:
_i+i
h2U
(h2Uv)
Azivi+1 + Azi+1vi
Azi + Azi+1
h+(vi+1 - vi)v i+1
-1
=
((hi+1 + hi)/2)
2
h2UvZ+i
(5.6)
(5.7)
-
Azi
h2 Uvz I
(5.8)
The artificial viscosity is added to dissipate the high frequency oscillations seen around
discontinuities and sharp gradients in second order central difference formulations.
The
basic idea is to replicate the viscous term of the momentum equation on the scale of the
grid (mesh for 2-dimensional cases).
The formulation is such that the N.V.T. induced
upwinding is applied locally near the discontinuities and becomes negligible in smooth
regions, [35, 31, 10].
5.1.2
Spatial discretization
The right hand side of the time marched equations , f(y) (Eq. 5.2), is discretized on a nonuniform grid. The grid has a higher concentration of nodes near the needle exit. The rationale is that the steeper gradients will occur near the needle's exit. Due to the non-uniformity
of the grid we use non-uniform second order finite difference formulas to approximate the
derivatives. For the first derivative
(Mi =
1
+
Azid+ Azi+1
( 2 "d
order central difference):
[ Azi
zi (fi+1 - fi) +
Azi+
Azi
+1_z__z_+
(fi - fi-)
1z
- R
6
6
one sided differences are used for end points. The forward difference is:
67
fzz
(5.9)
(fM)
=
SAZi+1 + AZi+2 (fi+1 -
Azi+2
Azi+ fi)
f
AZi+1
(fi+2- A)i) \
AZi+2 AZi+1+
AZi+ 26
+R (LAzi+i(Azi+1+
iAzi+ 2 )
(5.10)
For reference these formulas are taken from Hirsch [30], but they are derived from a
Taylor series expansion. Second derivatives are calculated by differentiating (with the same
schemes) the first derivatives.
5.2
Electric potential calculation
The second part of the problem requires us to calculate the electric potential, and electric
field intensities at the surface of the liquid which will vary as the surface charges and the
liquid jet shape fluctuate. As mentioned in chapter 4 this requires the solution of Laplace's
equation. A full two dimensional axisymmetric solution is one option, but it would require
mesh refinement near the jet surface to capture accurately the effect of the surface charge
density. The challenge here is that the jet shape and surface charge are fluctuating, and
therefore the mesh refinement would have to be adaptive and computationally expensive.
A second approach is to use a boundary element solver, as previously employed by Lozano
et al [48],Khayms [38], and Pantano et al [55].
This method yields only the fields and
potential at the surface of the liquid and needle, which is sufficient for our purposes. This
subset of equations is comprised by:
1. internal charge conservation 4.11
0 (27rKEth 2
27rhKE.
1 + cos a)(5.11)
cos a + z
2. potential equation as function of total charges
0 = f(z, h, En, E)
(5.12)
3. Tangential field equation (Laplace's equation)
dq$
dz
Et=-cosa-
68
(5.13)
adsj
a
'I
'I
II
Ii
I,
Ii
Ii
Ii
Il
'I
Ig
'I
A-
h,
h,
//
/
L
Figure 5-1: Depiction of the ring elements used to calculate the normal and tangential
electric fields on the surface of the jet.
The total charge (free + dipoles) per unit length in the jet and cone is discretized
into rings, and the joint potential of these rings, their images on a ground plane, and the
free charge on an equipotential feed tube, acting as the emitter is computed on points on
the liquid surface, thus yielding, by differentiation, the tangential field. As illustrated in
figure 5-1 the contribution of a ring of charge j to the potential at i is calculated by
oTds-h
4
=
a
=h
b
=2hihj
2 pr
dp
7T .7 o
+ h + (zi
1Z)2
(5.14)
acos W
(5.15)
(5.16)
The electrostatic boundary surface element solver has been modified from its original
form as presented by Lozano et al [48]. In Lozano's case the electrical charges were assumed
to be fully relaxed, so that o-T
-
e 0E.
In our case the liquid is not fully relaxed such that
aT = EO(E - E,). Expression 5.14 can be integrated in terms of complete elliptic integrals
of the first kind K(m), and the potential at node i can be expressed as a linear superposition
of those due to all the elements dzj as
6
4i = Nij (o,)
69
(5.17)
where
Nij
=
m
=
M-
=
iK(m) - VSi/K(mi)]
1[V
27rEO
6
zj
hi cos aj
h
4hih3
(hi + jhy)2
(5.18)
(5.19)
(Zi - Zj)2 +
4hih4(5.20)
(zi - zj + 2L)2+ (hi + hj)2
=
The ground electrode is simulated by adding a mirror image about z = L of every ring.
The images are denoted by (mi, K(mi)). The full matrix Nj is then partitioned into blocks
representing potential at liquid points due to charges on liquid elements (Ni), potential at
liquid points due to charges on the metal supply tube (Nit), etc. The liquid (#1) and needle
(0t) potential are expressed in matrix form by
-01(5.21)
Noticing that
#,
=
V is the applied voltage we obtain a single equation:
B
=
Nit N 1 V
(5.22)
A = (Nil-NtNtjNt
1)
(5.23)
=-B+A[co(Eno-E*')]
(5.24)
4
where the vector B comprises the effect of charges on the needle on the liquid potential
and the full matrix A the effect of liquid charges on the liquid potential [48]. The electrostatic solver neglects any external space charge effects and assumes that the non-neutral
and ambipolar layers of the liquid are thin.
The electrostatic equations 5.24,
5.13, and the conservation of internal charge 4.11,
are used to obtained the new spatial distributions of En, Et, and 0 for the updated jet
shape (h, a), velocity (u), and surface charge of the jet (a). This procedure is an analog
of the Poisson solving step in a Particle in Cell (PIC) codes. The new field and potential
distribution is calculated by using a modified internal charge conservation equation, [26]:
70
dE=
HEn
(5.25)
where the right hand side is the steady state internal charge conservation, Eq. 5.11:
HEn
= -E0
Eno + ECO59(21rKEtM(5.26)
1+ cosa
2rhK oz
/
The modified equation is time marched to steady state (dEn/dt -*
0) using an RK4
scheme. At each stage of the RK4 iteration equations 5.24 and 5.13 are used to obtain a
closed system of electrostatic equations. In more detail at each stage of the RK4 HEn is
recalculated using the value of Et from the previous stage. This HEn is used to obtain a
new En"* and subsequently o"*e
HEn
=
E,"**=
new)
E
where the coefficient
i
=
and E"
as shown next:
fCos a 0
a- - Eu +
21rhK 5z
n
60
E +
27KEth 2
1+ cos a
(5.27)
-A-HEn
=
B+A[Eo(En"ew - En)]
=
- cos a-(#
d
dz
(5.28)
(5.29)
e)
(5.30)
# changes for each of the i stages #
=
1.0, 2.0, 2.0, 1.0 where
1,2, 3, 4. These new values are used as updated predictors for the subsequent stages of
the solver. At the end of each complete loop the normal field is computed:
_
(Eno)m+l =(Eo)m"
Ate
+ 2HEn2 + 2HEn3 + HEn4 )
-(HEni
6
(5.31)
Once (E0)m+l is computed the potential 0m+1, and tangential Etm+1 fields are updated
as well from Equations 5.13, and 5.24. The iteration continues until the desired level of
convergence is achieved. After trial and error the non-dimensional "time step" and required
number of iterations were set to Ate = 5 x 10-5 and 5000 iterations.
In order to check that the effects on the liquid potential due to needle length are calculated correctly they are checked against those obtained with the Maxwell code, [53). This
independent check benchmarks the electrostatic subset.
The checks are carried out only
considering the needle and ground electrode setup, i.e. no liquid jet. The needle is set at
71
700
-simim
600 - 500
...
100
-
00
-
- --
-
50
z [I
ml
100
--
-
150
Figure 5-2: Comparison of numerical solutions for a finite length needle- electrode arrangement
is an
an operating potential V relative to the ground electrode. The Maxwell simulation
axisymmetric full 2-D simulation and the potential drop between the needle and electrode
is read at the symmetry axis. These results are compared to those obtained with our panel
method. For the case with no liquid charges the potential at the liquid surface is calculated
by:
01 = B = NitNJ 1 V
(5.32)
where the liquid surface charge is zero from equation 5.24. The comparison between
both codes is not direct, since one is read at the axis of symmetry and the other at the
liquid surface (effects due to liquid charges are ignored). The difference between the two
cases is minor due to the fact that the liquid jet is hj << ho Our results are shown in
figure 5-2. The results match each other very closely, the differences being of the order of
2-3 Volts at each location.
In figure 5-3 we can see the effect of increasing the aspect ratio R = Lt/(2ho) of the
conducting needle. These results clearly show that if the electrode configuration is modified
the potential distribution is affected dramatically. However, as the aspect ratio continues
of
increasing the effect tends to a limiting value (although the aspect ratio of the needle is
the order R ~ 1000).
72
IU(,
AR-4900
AAR
=2450
100 .......... .
0n
1
0.5
z [M]
-15
x10
Figure 5-3: Effect on needle-ground potential distribution for increasing needle aspect ratio
(Lt/2ho)
5.3
Integration of solvers
The two subsets described above are solved sequentially using the updated information of
one to update the other until steady state is reached.
In detail the time dependent equations ( 4.7, 4.9, and 4.12) are time marched for n steps
while holding the electrical fields constant (En and Et). After these n-steps, the electric
field is updated with the new values for radius (h), velocity (u and u'), and surface charge
(a-). The number of steps n < te/At where te = eo/K is the electrical relaxation time and
At is the time-step. A second criterion used to choose the time step is to ensure that the
C.F.L. condition is satisfied in order to preserve the stability of the Runge-Kutta scheme.
Typical values for At are of the order of 10- 9 sec. Trial and error were used to find a good
operating point.
For the electrostatic solver the number of iterations and its time-step where by trial and
error since the iteration is based on a fictitious time dependence.
5.3.1
Gridding
Ideally the grid used for discretizing the system of equations would be homogeneous, with
sufficient resolution to capture the cone to jet transition. Practical considerations, however,
discourage this approach. On one hand the first subset of equations (subsection 5.1) benefits
from having a fine grid that allows high definition of the transition zone. On the other
73
x 106
0~
0
0.2
0.4
0.6
0.8
1
1.2
z- axis [m]
1.4
1.6
2
1.8
x1
Figure 5-4: Jet radius vs z-axis for a typical simulation. Needle exit radius h0 = 6m
total length of liquid domain L = 150pam. The markers show the z-axis position of the grid
nodes, notice the higher density near the needle exit.
hand the electrostatic calculation (subsection 5.2) becomes increasingly expensive with fine
grids. For a grid with M nodes the electrostatic solver has to deal with a full matrix N with
M x M entries (Eq. 5.18), therefore, using a fine grid benefits the fluid part but makes the
electrostatic calculation much more expensive. From a physics point of view, most of the
action occurs at the transition region once the
jet
is formed there are no strong gradients
which suggests that there is no need for high resolution in the
jet
region. For these reasons
a non-uniform grid is employed. This grid has a high resolution near the cone-jet transition
region and a decreasing resolution in the
jet region, as seen in Figure 5-4 for a typical case.
For the most part the minimum grid spacing (Azi
=
zi1- zi) is set such that the
jet radius
h > Az.Typically for the high resolution section the spatial resolution should be about a
factor of 10 smaller than
5.3.2
jet radius (Az
mi[(h;et)/11).
Initial and boundary conditions
Initial conditions for this simulation were obtained by starting with a
and no voltage.
jet at high flow rate
The reason for this is that the initial conditions for the surface charge
density, as well as the electric fields are unknown. Alternatively analytical estimates may
be used as initial conditions. In our case we decided to use the first approach since it would
give us a gentler initial condition for each case.
Subsequent runs
lowered the flow rate
and increased the operating voltage. Each run used as initial conditions the data from the
74
previous converged solution, finally leading to the regime of interest.
Upstream boundary conditions are stipulated for the mass conservation, momentum,
and charge relaxation equations. As mentioned before the simulation reproduces the basic
setup of [13] with a conducting capillary needle facing a ground electrode at a distance
L. For the time dependent equations boundary conditions are given only for the upstream
end (needle exit). The initial jet radius is set equal to the needle inner radius (ho), the
velocity is set as uo = Q/(?rh'). The surface charge density boundary condition is specified
as described in section 4.6. The applied operating voltage is given by equation 3.15
5.3.3
Residue calculation
The convergence of the simulation is monitored by keeping track of the residues for each of
the subsets. For the time dependent equations the residues are defined for the radius (h),
velocity (u), and surface charge per unit length (uh):
ma hn+1 -- h
RESh
=
max
RESu
=
max
RESgh
=
max
(
n+1
(ohn1-
A-
)(5.33)
(-3
)
(5.34)
(u-h)0
)
(5.35)
Theoretically as time approaches infinity the residues should approach zero for a stable scheme with solutions. In practice the degree of convergence is dictated by practical
considerations such as an acceptable variation of the solution.
As mentioned above the potential solver is run every n number of steps to update the
E-field and potential distribution. For the potential solver the residues are:
RESEn
=
max (En+' - En)
(5.36)
RESEt
=
max (Etn+1 - Et)
(5.37)
RES4
=
max (On+1 _ O)
(5.38)
In this case, it is also checked that the fluctuations of the variables tend to zero. other
75
parameters that are checked for convergence are the total current which should be constant
along the jet (dI/dz = 0), the calculated flow rate at every axial location should also be
constant and equal to the one specified as upstream boundary condition.
Through post-
processing an energy breakdown is made in order to check that energy has been conserved
(see section 6.2.3).
5.4
Summary
In this chapter the numerical schemes used to solve the system of numerical equations
developed in chapter 4 have been presented. The equations are separated in two subsets:
time dependent electrohydrodynamics and electrostatic sets. The details of the numerical
scheme used in each case have been presented along with their corresponding initial and
boundary conditions. Finally, the integration of both solvers has been presented.
76
Chapter 6
Results for droplets
6.1
Results and discussion
In the following, we present a comparison of our numerical results for emitted current to
experimental data from Fernaindez de la Mora & Loscertales [13]. Our numerical simulations
have validated Fernandez de la Mora current vs. flow equation to within experimental
accuracy, and have also loosely confirmed his empirical minimum flow condition. But in
addition, they have illuminated the inner workings of the electrospray phenomenon and
have uncovered subtle variations on the general theme. Some of these are in retrospect,
understandable from simple basic balances, others remain obscure. We attempt to combine
numerical results and observations with basic principles to see if a clear pattern can be
discovered.
6.1.1
Reference experiments
Fernandez de la Mora and Loscertales experimental data were obtained using a fairly standard electrospray setup where the meniscus emerged from the end of a stainless steel capillary needle facing a perpendicular flat ground electrode (9 mm away from the needle, in
most cases). Two sizes of needles were used: one with 1.07 mm OD and 0.81 mm ID, and the
other with 0.41 mm and 0.24 mm respectively. The experiments were conducted at laboratory pressure and temperature (and in the case of H20 with a CO 2 atmosphere to prevent
breakdown). Current measurements were made with an electrometer connected between
the collector electrode and ground. The flow rate was controlled by an imposed pressure
77
100
90
FM
W
80
V 1-Oct
*
BA
0 FM1 simm
70
v
o
>
*
-a
S50
FM2 simm
W simm
1-Oct simm
BA simm
E
VV
40-
V.3
30
20 -
20
10 0
0
E
*
-
>
1
V
t
4
4
3
2
5
6
Figure 6-1: Dimensionless spray current ( [Eq. 3.51] versus 7 [Eq. 3.50] comparison between
experimental (shaded symbols) and numerical data(unshaded symbols). Experimental data
from [13].Numerical data based on various liquids (Table 6.1) and flow rates (7 > 1).
difference
AP and was inferred in many cases from the linear relation AP/Q obtained
experimentally at higher Q (originally calibrated by the moving bubble front method).
6.1.2
Numerical results and Comparison
Figure 6-1 shows a comparison of the numerically computed results versus experimental
data using 7 and (, from equations 3.50 and 3.51. The properties of the liquids employed
in the simulations can be seen in Table 6.1.
It can be seen that the numerically com-
puted currents compare quite well to the experimentally observed values of Fernindez de
la Mora & Loscertales, [13]. The empirically determined proportionality factor introduced
by Fernindez de la Mora, f(E) is the slope of each data line in Figure 6-1.
Figure 6-2 shows three sets of data. The first set are experimental values for f(E)
from Fernindez de la Mora and Loscertales, [13].
is from Chen & Pui, [4].
The second experimental set of data
The third set of data are our computed values for
f(e)
for the
fluids of Table 6.1. The simulations show good agreement across the Fernindez de la Mora
experimental range. Chen and Pui suggest that the deviations they see at high permittivities
E > 80 relative to Fernandez de la Mora and Loscertales may be due to the fact that they
78
liquid
K [Si/m
E
y[N/mj
p[Kg/m 3]
p[Pa- s]
3
f(E) (experimental)
Formamide-1
Formamide-2
111
111
0.00100
0.01000
0.058
0.058
1130
1130
3.80x103.80x10- 3
18
18
Water
Benzyl Alcohol
80
13.1
0.00215
0.00790
0.073
0.040
998
1040
1.00x10- 3
7.76x10-3
18
8.5
1-Octanol
Ethylene- Glycol
10.34
38.66
0.00116
0.01690
0.026
0.048
825
1100
1.06x10- 2
21x10- 3
7
17
Table 6.1: Properties of liquids used in the simulations from [13]
employed different doping agents to increase the conductivity of their solutions.
Only our simulation results for Ethylene Glycol show an important difference relative
to experimental data. Three cases were run for various flow rates and in every case the
total current was well described by expression 3.43. The value for f(e) as computed was
13 instead of 17 as given by Fernandez de la Mora.
Previous studies by GaiAn-Calvo
and coworkers have produced current (I) vs flow rate (Q) scalings which are independent
(or nearly so) of the liquid relative permittivity, [15, 18, 16, 17]. Gaiian-Calvo's scaling
(Eq. 3.47) suggests that
(E) in Eq.
f(c)
,
i.e. cancelling out the effect of the relative permittivity
3.45. It is noteworthy that a simple fit to our results of the form f(E)
2V
would be fairly accurate. This is in closer agreement to the scaling of Gafiin-Calvo obtained
with a hybrid experimental-numerical technique. In that work
f(c)
= 26V,
[16]. From
equation 3.44, this indicates that the current has very weak dependence on the dielectric
constant. We will later see from detailed computational results that the surface charge o- is
in all cases within 20% of its equilibrium value eoEno, even in the neck region. If the surface
charge were everywhere fully relaxed, the dielectric constant e would not play any role in
the charge and fluid dynamics, and our results for the current appear to come close to this
limit.
6.2
Sample results and parameter studies
The numerical results shown next were obtained using Formamide (K=0.005 Si/m) with
a needle (capillary tube) radius of ho = 10pim, the collector was set at L = 300pm, and
the applied voltage was computed from the startup formula (Eq. 3.15), V = 1150V . Two
cases are shown in Figure 6-3 and 6-4; the first one (solid line) with 77 = 2.2 and the second
79
25-
Chen-Pu
FMI,FM2
20----
-
15'H20|
15 S
10 3E0
10-BA ,
BA -
00
20
-
-
,'
''
EdeM
EG
2(9)/2
60
40
e relative permittivity
80
100
120
Figure 6-2: Comparison of experimental data from [13] and [4] to numerical results for
various dielectric permittivities,(E) versus non-dimensional slope f = (/7
one (dash-dot line) at q = 1.6.
The plots show the jet radius (h), mean velocity (u),
surface charge density (a), liquid potential (<01), normal outside Electric field (E*) and the
tangential electric field (Et) as one progresses from the needle to the collector.
These results (Fig. 6-4) show that as 77
-
1 the transition region becomes sharper with
the cone semi-angle (a) approaching the Taylor cone value (aT = 49.30), and the emitted
jet becoming thinner. For a lower 77 the jet average velocity (u) increases significantly while
its profile remains almost constant as shown by the very low values of the shear velocity,
U', (Eq. 4.6, the difference between the surface and centerline velocity). The surface charge
density in the jet is seen to decrease as 7
-
1. It can be seen in both cases that the potential
in the cone region remains almost constant. This nearly equipotential condition at the cone
is similar to the behavior found in Taylor cones. Once the cone becomes a jet the potential
changes rapidly until it reaches zero at the collector. The electric field distribution reaches
a maximum somewhere in the cone-jet transition region. In all cases it is observed that the
normal component is stronger than the tangential one. Notice that the normal component
of the E-field increases (although weakly) as 7 decreases. The tangential component shows
the same behavior but the change is more dramatic. As the liquid progresses into the jet
section the fields slowly decrease in magnitude. Finally, the last subplot shows the spatial
distribution of the total current (I) and its components: the conduction Icii and convection
80
3
1
10
1
x 10
-
1.5
0.8
1
100
0.6
0.8
0.6
T1=2.2
0.4
0.4
60.5
50'
0.2
02
0
0
1
0
0
2
[ml
x
0,
0
2
1
[m]
x
15 X l0
1500
5 x10
10-
-4 1000'
1
[M]
2
X10
1
[m]
2X10-4
0
1
2
[M]
x 10
7
4
2\
3
1.5
0
0
0
X104
7
2
5
500
0
1074
0.5 11
00
1
[ml
2 4
x i0
0
0
1
[M]
2
X 10-4
0.5
1
[M]
1.5
2
X 10-4
Figure 6-3: Simulation results for FM, K = 0.005 Si/m ho = 10 Am, L = 300 pm and
applied voltage V = 1150 V. Two flow rates shown 77 = 1.6 (dash-dot line) and ? = 2.2
(solid line)
current I,.
The total current distribution (I = Icd + I,,) remains constant (i.e. dI/dz
=0) which means that charge is conserved. Secondly, the total current decreases with 77 as
expected (although not shown the computed current is in good agreement with the scales:
Eq. 3.45 and Eq. 3.47). Thirdly, it can be seen that the conduction current dominates in the
cone section and that the convection current dominates in the jet section. The point where
both current components become equal is near the cone-jet transition, a few jet diameters
into the jet.
6.2.1
Electrode distance effect
It should be noted that as the electrode is positioned further away the required starting
voltage will increase as given by expression 3.15. As mentioned before this expression is only
approximate and may be off by about 10% giving rise to slightly different total currents.
The effect of varying the operating voltage on total current, at a given flow rate, is addressed
in detail in section 6.2.7. One lesson learned from this set of numerical data is that in order
to faithfully reproduce results the geometrical configuration of the needle-electrode needs
to be well documented as well as the starting and operating voltages at which experimental
81
X 105
0.4
-
__
... --.. --
-..-..-...
___
I
0.2---------0
0.5
1
z-axis [m]
1.5
2
2.5
x 10
Figure 6-4: Close up of the cone-jet transition of figure 6-3, K = 0.005 Si/m h 0 = 10 pm,
L = 300 pm and applied voltage V = 1150 V. Two flow rates shown i = 1.6 (dash-dot
line) and 71 = 2.2 (solid line)
results are obtained.
Computations for various electrode distances (L = 150,300, 600ptm) were carried out to
characterize the effect of the ground electrode boundary condition (i.e.
#
= 0 at collector).
The results in figure 6-5 show (a) that the sudden increase of the field intensity at the end
of the computational domain is simply an artifact of the electrostatic ground boundary
condition (b) that for lower conductivity liquids the residual conduction current may be
affected by the electrode.
The minimum electrode distance was chosen based on the aspect ratio L/ho reported
by [13]. In his case the aspect ratio was no less than 18. In our case the smallest aspect
ratio was 15 up to, in some cases, 100.
A second numerical experiment is to fix the operating voltage to a given value and run
the simulation for the three previously mentioned electrode distances. The results for this
case are shown in figure 6-6. In this case the operating voltage was set to 1190 volts. In
this case, however, the total currents are almost the same. This set of runs confirms that
the spurious end effects at the electrode do not introduce significant errors in the important
cone-jet transition region.
82
6x1610
-
8x 10
~
6
4
0
0
0
3
0
1400
U
5
-1200
1000
4
*~800'
R.600
*0
.0
0~
400
0
0
0
1
0
4
3
2
[mI
4
ml
-
4
06
XI
7
2
10 X 107
,
4
6
200
0
0
xLU
4
1.4
10
1
0.8
0.6
0.4
5
6
x 10
07
15
2
0
2
4
0.2
2 in]
0
00
4
0
2 [iM] 4
0
2
4
[rn]
10-4
x 104x
x 1
Figure 6-5: Simulation results for FM1, ho = 6pm, 7 = 1.26. Electrode distance is L =
150pm, L = 300pm, and L = 600pm, with applied voltages V = 911V, V = 1050V, and
V = 1188V respectively.
xx10,
8 x1400
6
571200
5
-
6 '.1000
.- - ..
- - - - - ------
S2
4800
600
,
2400
200
0
0
0
1
2
[in]
7
3
0 .-4
1
S0
2
x 10
ml
x10
10
x.
3
0
2
4
m7
x 10
10-x
6
x
1
10
4
l05
02
1
s'N0.4
0.6
0
-
0.2
00
0.
0
1
2
[m
3
x 10-
0
1
2
3
x 10
0
-
1
2
3
X 10
Figure 6-6: Simulation results for FM1, ho = 6pm, q) = 1.26. Electrode distance is L
150pm, L = 300pm, and L = 600pm, with applied voltages V = 1190V.
83
=
x 10
4
computed E.
E
3
2
.-
- -
1~
0
x10
51
z [m]
4
E3
Et
2.
computed Et
0
2
I
0.5
1
1.5
4
[m]
106z
1.5-0.50
0
0.5
1
1.5
z [m]
x 10-4
Figure 6-7: Comparison of analytical Estimates for Electric fields and jet radius to simulation data. Liquid data: FM2 at ? = 1.26, h0 = 2pm, L = 150Mm
6.2.2
Analytical estimates compared to numerical results
In this section analytical estimates for the electrical fields of the jet and cone sections are
presented. The present analysis assumes that the liquid charges are electrically relaxed;
the numerical results show that this is true, with the possible exception of the cone-jet
transition region.
Previous plots have shown that as the non-dimensional parameter 7 -- 1 the angle of the
cone region tends towards the Taylor's angle (i.e. at
=
49.290) and the cone region tends to
an equipotential. This similarity to the conditions upon which the Taylor's analysis is based
suggests that the outside normal electric field, (En) can be approximated by an equilibrium
between the electric and surface tension stresses, leading to
E'
Enc
=
V
2ycosa
60oh
(6.1)
Downstream from the transition zone most of the charge is convected on the surface
allowing us to approximate the normal outside field by using Gauss law and noting that
dq/dx ~ I/u, [21]
84
E.
hI
2QEo
=
(6.2)
The tangential field in the cone region is estimated by using equation 4.22 and Id, =
I - I, where I is given by 3.44 and I, by equation 4.21 for a fully relaxed cone (i.e.
0- = 1EE). The result is
c=
f()h
1 yQ \
(EK)
2Q
1/2
rh3K
22yo
h)
1/2
(6.3)
The tangential field in the jet section is dominated by the charge from the cone region,.
A first approximation to the tangential field distribution can be computed by evaluating
the Taylor's cone field solution (cone with apex semiangle aT = 49.3* and no jet at 9
=
7r,
i.e. at the z-axis), [15, 61.
CT
Et
=Q
1 12 (COSir)
aT)
/2 tan a' Q1/2(cos
= 0.67
(6.5)
C=
TQy )2
Eoz)/
-
(6.4)
where Q1/2 is the Legendre function of the second kind, with degree 1/2 and order 0;
while Q'1 /2 is its derivative with respect to 9.
Finally the jet radius dependency given by Gafian-Calvo is a function of order h
C
O(Q 3/ 4J-1/ 4 z-1/ 8 ), [15]. This estimate assumes that the jet is fully relaxed (- = EoE*) and
that the normal electric field is well approximated by equation 6.2. The main assumption
for this estimate is that in the jet region the momentum balance is given by:
1du2
2dz
_=2aEt(
ph
which after integration gives the form:
e-
1
(21r2CT)1/4
p2
Q1/8Q
/
3 4
1/41/8
(6.7)
A comparison between these estimates and simulation data can be seen in Figure 67. The simulation data (solid line) shown is for FM2 at q1 = 1.26. It can be seen that
the estimates give a reasonable approximation to the numerically computed results. An
85
exception is Etj, which is overestimated by (6.5). Clearly, the jet charges do provide a fair
amount of space charge and reduce the axial fields due to the cone alone.
Previously, in figure 6-3 it was observed that as 7
---
1 the surface charge density
decreased. Using the analytical estimates, this behavior can be explained qualitatively by
noting that in the jet I,
I and therefore o ~ 7rhI/Q where (h) is approximated by
equation 6.7. This shows that if h oc (Q5 /8 ), and I oc (Q'/ 2 )(see equation 3.44), then the
surface charge density, a oc Q1/8
x
of the surface charge density a ~
rhI./Q where only h is function of z, (equation 6.7).
1/4.
Notice that in terms of the spatial variation
The surface charge density then can be seen to vary approximately as a oc z- 1/ 8 . In the
following sections these analytical approximations will be used to get a better grasp of the
underlying physical phenomena observed in various numerical parameter studies.
6.2.3
Energy Conservation
We calculate the power input of the system by multiplying the extraction voltage by the
total resulting current. This power is used for accelerating the flow, setting up the cone-jet
shape (surface formation), and some of it is dissipated by viscous stresses, and Joule heating.
In our case we present the energy analysis in terms of voltage V = Power/I and energy
percentage (V/Vt, V8t from Eq. 3.15).
The sum of all the voltage contributions should
ideally be the original extraction voltage (Vg). Detailed numerical results for Formamide
(K = 0.01 Si/m) vs 7 are shown in figure 6-8. In order of importance the different energy
contributions are given below. The acceleration voltage (kinetic energy) is given
Vk =
2--=L
(6.8)
where u is velocity at z = L. This voltage is used to accelerate the fluid to its final
velocity. The second contribution to the voltage is due to Joule heating. This voltage is
calculated as
KIL 27r(hEt)2 d
1JL 1+ cos(a)
The third contribution is the energy needed to overcome surface tension and set the
shape of the liquid jet. The expression for this voltage is:
86
lUU
Idne
0ohmic
-
--
90 v surface-
10 viscous ..... .
-.
8 0 -..........
800 -.-.
. ...
......
..
.. .. .
- --.
.........
......... -- --- -...--.--..-.........
--
-.-- - -.--- -. - --.........
.....
-.-.-...--- - -- - -....
----. .. -.
-. ....
7 0 -- ....
-
--. -.--.
6 0 -..........
......... - - - -.-.--.--...
.........
.. .........
---.. -...-..-....
--.
-.
5 0 -- ....
VV,
-...
----. ....
40 -- ....
70
--- - -- -
-...
-.-.--.--.--......
.........
- -- - - --.-.-
.
..
-....-..
-.
VV
U...
- -- - - -- - ..--
-- -
..
0)1
1 0 ..--..
1.5
1
---.-.
---..-
--..-.--.2
2.5
3
Figure 6-8: Energy breakdown (percentages) vs q for Formamide, K =
0.01 Si/m
Vy = 0r)(hu)|z=L
The remaining contribution is that due to viscous dissipation given by:
3 fp L
* V=r 7I
0
0
aU
--z
2
dz.
(.11)
Adding up all the different contributions the applied potential should be recovered. In
reality a small percentage, in the order of
1- 3% of the applied potential is unaccounted for.
This "lost" energy may be due to unaccounted smaller
losses.
Figure 6-8 presents the voltage contributions (normalized by the starting voltage ,V
vs q for a formamide solution
(FM2). In general the dominating contributions (for the
liquids in this study) are the kinetic energy and the ohmic heating. The surface formation
energy
(surface tension term) varies between 5-10% and viscous losses amount to 1-3%.
Figure 6-8 shows that asy
increases.
increases the kinetic fraction decreases while the ohmic fraction
The contributions due to surface formation
almost constant, which makes sens ede
(surface tension related) remain
basic shape of the cone and jet structure will
not change dramatically with
o.
Numerical results for the ohmic dissipation fraction V/V (formula 6.9) vs
87
for various
50
0
o
l
45-
Fml
Fm2
water
1-Octanol
Benzyl-alcohol
4
*
40-
2Cu 35
300)
E
9-
0
2520-
0
15
I
4
0
0 Co
4
01'
1.5
*
* *
.4
2
2.5
3
4
3.5
4.5
5
Figure 6-9: q vs Ohmic dissipation fraction (Vn/V) for multiple liquids.
VO/v
100.00% 00.00%
______
IA
80.00% -
___
mEG____
70.00%60.00%
_
00.00%40.00%___
30.00%
20.00%
__
__
40.00%
60.00%
10.00%0.00%
10.00%
20.00%
30.00%
60.00%
70.00%
80.00%
90.00%
100.00%
VestN
Figure 6-10: Plot of Ohmic dissipation for various liquids. x-axis estimate based on equation 6.12 vs simulation data
88
liquids are shown in figure 6-9. The numerical results show that as 1 increases the ohmic
dissipation fraction increases as well. The trend shown in figure 6-9 is that the liquids with
the slowest electrical relaxation time (EEo/K) present higher levels of ohmic dissipation. This
can be seen by noticing that in figure 6-9 FM1 has the slowest relaxation time (te = 0.98ps),
and highest ohmic dissipation, vs Benzyl-alcohol which has the fastest relaxation time (66
times faster than FM1), and lowest ohmic dissipation.
A normalized estimate for the energy dissipated as ohmic heating can be obtained by
integrating expression 6.9 divided by the total applied voltage 3.15.
The integration is
carried out using estimates 6.5, and 6.7 from the beginning of the jet region zo = FRd
(where it is assumed F ; 2.3) to z - oo
-
V
=
2
E
=F 1/4
-R
\f(E)
d-
(6.1 2 )
2
ho ln(L)
A comparison of the percentage of energy dissipated as ohmic heating between simulation
results and equation 6.12 is shown in Figure 6.2.3. The plot shows that expression 6.12
captures the right trends but is not a very accurate estimate. Reasons for this inaccuracy
may be traced back to the uncertainty of the transition point zo and also to the fact that
expression 6.5 neglects the effects of charges on the jet.
6.2.4
Cone-Jet transition length scales
Previous research has aimed to identify the length scale that best describes the transition
from the cone to the jet region. Fernandez de la Mora suggested that the jet could scale
with the electrical relaxation length r* (3.16), or the inertial length Rd (3.19). Other studies
by Gafi'n-Calvo led to the introduction of another length scale, Rg (3.26), [151.
In this case the transition has been defined as the point where the convection current
equals the conduction current. Previous work by Gafian-Calvo [15] and numerical results by
Higuera [28] have shown that I,,
=
Ica occurs somewhat downstream of the point of maxi-
mum curvature. Nonetheless, choosing the crossover point (R.)
results in a non-ambiguous
definition of the cone-jet transition and it serves as a lower bound of the transition radius.
Numerical results for various liquids were used to further investigate how the transition
region relates to Rd, r*, and Rg.
The results are shown in figure 6-11 where the plots
show on the horizontal axis the length scales Rd (top), r* (center) and, Rg (bottom) in
89
4
SFMI
* FM2
o H20
Oct
N
*Alc
3
0
0
I
0
I~
1)
3
1
2
3
4
r [± m]
5
6
7
8
16
18
.FM1
$ FM2
o
*
*
H20
Oct
Alc
3
0
I
4$
2
4
6
8
R,
10
[p m)
12
14
4
3.5
3
* FM2
0 H20
1 Oct
* Alc
0
2.5
2
1.5
0.5
0
0.5
1
1.5
2
2.5
R [A mI
Figure 6-11: Cone to jet transition radius (R.) vs proposed length scales.
3.16), center Rd (Eq. 3.19) , bottom R,
90
top r* (Eq.
0.90.8 0.70.60.5 0.4*
0.3 -
V
0.20.1-
a
PC
0
3
2.5
2
1.5
1
0.5
3.5
4
6
7
r[g m]
0.80.70.60.5
0.4*
0.30.20.1
C
0O
4
3
2
1
5
R [AM]
0.90.80.70.60.50.4-
*
*
v
0.3-
0.20.10
0
VV
0&*
0.5
1
R [V m]
1.5
Figure 6-12: Breakup radius for TBP (various conductivities) (Rb) vs proposed length scales.
top r* (Eq. 3.16), center Rd (Eq. 3.19) , bottom Rg. Experimental data reproduced from
Gamero and Hruby [22].
91
x 10
I50
X-
2
EE
0.8
1.5
00
0.6
0.8
Si/rt,-'_
K=0.005
-
0.4
50
0.2
0
1
0
2
[m]
X10-4
0
0
- 0.4.
K=0.001 Si/
1
[M]
0.5
1
0
2
[m]
0
0
X 10-4
1
0
2
[m
y4
x
2X 10'
1.5
-
'cv
3
2
0
2
X10-4
4
5
-,
1
5 x 10
10'
500
0.
0.2
[m
- ..
41000,.
.
0
2
x 10~4
15 X 16,
1500
0.6
1
1
[m]
0
0
2 4
X 10-
0.5
I.
1
[m]
2
X 104
0
0
1
[m]
2
1X4
x
Figure 6-13: Formamide data q = 1.6, ho = 10 im, L = 300 Am and applied voltage
V = 1150 V. Two conductivities shown: Solid line K=0.001 Si/m, and dash-dot line
K=0.005 Si/m
pm and on the vertical axis the crossover radius R, also in um. It can be seen from the
plots that for the case of R 9 most of the data collapse on one line. A linear regression for
this case gives R 2
=
0.9933 with a slope of 1.45. Comparing these results to those obtained
experimentally by Gamero and Hruby [221 we show the same qualitative behavior for r*, Rd,
and Rg vs their calculated breakup radius Rb (see figure 6-12). Gamero and Hruby report
that Rb ~ 0.6Rg (coinciding with the slope value given by Gafiain-Calvo [15]) whereas for
our numerical transition results indicate R,
~ 1. 4 5Rg. From the experimental data the
better correlation to the breakup radius (Rb) is given by Rg, [22]. The same behavior is
confirmed from our numerical results for the transition radius (R,).
6.2.5
Conductivity effects
A set of results is shown in figure 6-13 for two conductivities, namely Formamide K
0.005 Si/m and K
L = 300
=
=
0.001 Si/m. These results were obtained with 7 = 1.6, ho = 10 Am,
rm and applied voltage V = 1150 V. Note the important differences between the
two cases even though their non-dimensional slope f(E) (and hence current) is about the
92
x 10~~
0.
K.5O11.
2
2.5
z-axis [m]
x~1o0
Figure 6-14: Close up of the cone-jet transition of figure 6-13. Formarnide data y = 1.6,
40= 10 ,im, L = 300 ,im and applied voltage V = 1150 V. Two conductivities shown:
Solid line K=0.001 Si/rn, and dash-dot line K=0.005 Si/rn
same (5% difference). The cases were run at the same y = 1.6 and both cases coincide with
the current predicted by Fern&ndez de la Mora from equation 3.44. It should be pointed
out that in order to keep the same
an order of magnitude (i.e. y, c
1
for the two liquids the flow rate was decreased by
ty
/KQ) for the K = 0.005 Si/rn data. Figure 6-13 shows
an important change on the residual conduction current Icd in the jet region as a function
of liquid conductivity. The ratio of conduction current to total current in the jet can be
estimated by :
(6.13)
-e ~Eh
X
lI
approximating the jet tangential field by 6.5 and the jet radius by 6.7 we obtain :
Is
E 33/2 F1/4E1/2 1 3/2
K1/2p1/4J z/
2 \ f(e)/
CT
I
/
(6.14)
This estimate shows the residual conduction current fraction is inversely proportional
to square root of the conductivity, which corresponds to the trend shown in the numerical
results of figure 6-13. Estimates using expression 6.14 give a conduction fraction of 0.071
(numerical result 0.077) for the K = 0.O01Si/m solution and 0.032 for the K = 0.005 Si/rn
solution (numerical results give 0.018) at z = 250 tim. Equation 6.14 also indicates the
current fraction dependence to the liquid relative permittivity Iai/I Ee/
93
4
(where f(E) a
X10"s
7X
Formam8de
8
1
10,
E.
1=4+4d
0.8
6
=5
0.4
M
44-
3
Taylorangle
.2
0.2
2
2
0
0.5
1
In4
1x1
l O
110"
0
1.5
0.5
1
X16
x107
actanal
1.5
0
5
10
[m
xO10
15
x 10
5
8X 101
Eag
.
10 z
0.8
]
[
I
77\
-6
E6
5
0.6
322
0.2
42
00
0.5
1
[m
1.5
X1
2
4
0
05
1
.5
xO
0m[
0
4
0.5
1
[m]
1.5
X10 4
Figure 6-15: Formamide-1 (e = 111), and Octanol data (E= 10.34)
,r = 1.96
.
Both cases with
Vef). We have verified numerically that this fraction is indeed higher for FM (e = 111) than
1-Octanol (E = 10.34), see figure 6-15.
6.2.6
Dielectric permittivity effects
Figure 6-2 shows two distinct behaviors, one for high dielectric permittivities (e > 40) and
the second for the lower permittivities. Two runs were conducted for Formamide-1 and 1Octanol both fluids at different extremes of the Fern6ndez de la Mora's experimental range.
The results for the final distributions of jet shape, normal electrical field, charge relaxation
and current are shown in Figure 6-15. One of the important things to notice while comparing
the jet shape for both fluids is that for the Octanol case the cone angle almost coincides with
Taylor's cone angle for a conducting liquid (a = 49.30). In this case the cone region becomes
almost equipotential thus approximating Taylor's exact solution.
relaxation (i.e. En -++-/Eo)
The degree of charge
can be seen to have a strong correlation with the remaining
fraction of conduction current in the jet region. The residual conduction current fraction is
smaller for the 1-Octanol case (almost completely relaxed) as opposed to the Formamide1 case. The importance of the relative permittivity for the conduction current fraction
94
25x 104
2.5X
2.42.32.2S2.1-
S21.9
-
..-
-----
1.81.7-
1.61.5
1250
1300
1350
1400
Voltage [V]
1450
Figure 6-16: Total current (I = Id + I,,) vs Voltage for water, 7
300pum, original starting voltage V = 1372 V.
1500
=
1.55, ho = 10pim, L
=
can be seen from equation 6.14. The conduction current fraction is directly proportional
to (/f(f)) 3/ 2 which is larger for Formamide-1 than for 1-Octanol. Expression 6.14 gives
a conduction current fraction of 0.19 for FM1 (the computation shows 'cd/I
~ 0.19 at
z = 1x10- 4 m) and of 0.04 for 1-Octanol (computation shows Ica/I ~ 0.019at z = 1x10- 4 m).
6.2.7
Voltage parameter study
The simple theories based on a cone with Taylor's angle appear to indicate that the singlecone regime is attained for a given liquid and flow rate only at a particular voltage (V),
somewhat lower than the starting voltage given by 3.15.
Experimentally however, it is
known that the single cone regime can be maintained within a narrow voltage window for
a given configuration. The voltage window usually spans between 10% below the minimum
extraction voltage and almost 40% above. A parameter study has been carried out to explore
the effect of increasing the extraction voltage. The liquid used was water, ho = 10 Am and
L = 300pim. All runs were carried out at q = 1.55.
The current plots can be seen in figure 6-16. The reference starting voltage (V=1372 V)
95
x 10'7
I=IC
+IA1472V
71.5
1322 V
1422 V
0.5-
t
0
1222V
AIc
-
z-axis
[ml]x
2
3
Figure 6-17: Convection (I,) and Conduction (Icj) current vs Voltage behavior for water.
7= 1.55, h0 = 1Opm, and L = 3OpOm. Reference starting voltage 1372 V
x
10~
0.91222V
\.
0.8
..i\
.0
0.7
1322 V
0.6-
%%%
0.5
-%
0.4
.1472
0.3
--
1422 V
V
0.2-0.1 a~t=49.30-'~=
0
0.5
-=--
1
1.5
2
z-axis
[m]
2.5
3
3.5
x
10~
Figure 6-18: Effect on jet shape at different voltages. Liquid: Water at 7
and L = 300pm. Reference starting voltage 1372 V
96
=
1.55, h 0
=
1Opm,
4:00
Single Cone
Multiple Cone
350030002500G
200015001000500
2000
2500
3000
3500
4000
Needle voltage [V]
4500
5000
55D0
Figure 6-19: Experimental data from Gamero-Castaflo [52] for glycerol. Needle h 0 = 50pm,
L = 4mm and fixed flow rate Qo = 12.5nl/s). Notice the change from single cone-jet mode
to the multiple cone-jet mode (highly stressed regime)
was calculated using equation 3.15. Results show that the overall current does have a slight
(approximately linear) dependence on the applied voltage. This almost linear dependence
to the voltage has also been numerically calculated by Hartman et al for ethylene glycol
solutions, [27].
The convection current Ic. (see figure 6-17) shows the strongest variation, increasing
with applied voltage V, whereas the conduction current (Ia)shows no important changes.
A related effect of increasing the voltage is a deformation of the liquid cone surface. Figure 618 shows this effect on the cone-jet shape for the voltages shown in figure 6-16. A qualitative
explanation for this effect is that the higher voltage augments the electric stress which in
turn is balanced by the surface tension stress (6P oc -y/R), (although a reduction in the
liquid pressure is also present). The surface stress equilibrium is thus maintained by a
steepening of the cone angle (i.e. reduction of the liquid curvature), see figure 6-18. Similar
trends of the cone semi-angle vs applied voltage have been obtained before for the case of
conducting liquid menisci subjected to an applied voltage, [62] [55]
Experimental data from Gamero-Castafno [52] for glycerol (needle ho = 50pAm, L = 4mm
and fixed flow rate Qo = 12.5nl/s) also showed an initial linear behavior from the onset
97
voltage 2400 V to 3230 V. Pictures taken during the tests also showed a shrinking of the cone
as the voltage increased. A sudden current increase was measured beyond this voltage and
observations confirmed the bifurcation of the single cone-jet to a multi cone-jet configuration
(highly stressed regime).
6.3
Summary
Numerical results for multiple liquids and flow rates have been presented. The predicted
current and non-dimensional slope compare quite well with published experimental data of
Fernindez de la Mora & Loscertales [13]. Our results reproduce the effect of the dielectric
permittivity on the non-dimensional current vs.
square root of flow slope for different
liquids as seen in the above mentioned paper. Further examination of our results shows
that (except near the minimum flow) there still exists a finite conduction current in the jet
downstream of the transition region. The conduction current fraction has been shown to
be related to the degree of electrical relaxation that the liquid achieves, although no causal
link has been identified.
98
Chapter 7
Results for mixed ion-droplet
regime
In this chapter initial results for field enhanced ion evaporation axe presented. As mentioned
in chapter 3 ion evaporation becomes an important phenomenon only for high conductivity
solutions (K - 1 Si/m). The maximum conductivity for which converged solutions have
been obtained is K=0.07 Si/m which is well below the expected range. Following, the work
of Higuera [29] the first approach has been to lower the solvation energy of the ions. The
first set of results are shown in subsection 7.2. The concern with this approach is that the
results are qualitative at best. There is no guarantee that this modification is physically
sound or that it accurately predicts the mixed ion-drop regime.
7.1
Expected results
The results obtained with the ion evaporation model are shown next. The expected behavior
is that seen experimentally by Gamero,[20, 47], see figure 7-1. The x-axis has the nondimensional flow rate 71 and the y-axis shows the emitted current, It&, with the arrow
pointing in the direction of increasing liquid conductivity, K. Notice that for the higher
conductivities (K ~ 1 Si/m) as the flow rate (77) decreases the emitted current (Itot)
reaches a minimum and then starts to increase again. It is in these cases when the current
due to evaporated ions becomes of the order of the droplet current. The contribution of
the accumulated ion current (Ii) to the total current (Itot) as function of flow rate (Q)
is easier to appreciate in figure 7-2 from Lozano, again for a formamide doped with NaI
99
500.00.
-*-F07
-F11
400.00-
F14
-
400.00
350.00
-
-11-F16
F22
-
FDM
-
-
---
-.--
300.00-
I =IE) 1 (wwP
200,00
50.00
0.00
--
-0
0.00
0.50
1.50
1.00
2.00
2.50
3.00
3.50
4.00
4.50
Figure 7-1: Fig. 11 from Gamero [20]. Current vs non-dimensional flow rate for formamide
solutions of various conductivities. The arrow points to increasing conductivity. The minimum is associated with onset of ion evaporation. F07 (K=0.71 Si/m) through F22 (K=
2.20 Si/m)
1.......
o
0
s
....
0.2
O
04
0.
06
.
07
Figure
ho
"2 current vs. low rate. formumkle + Nnl Ihe dahed liue
correopooda to the valk of the moo-dimensiona flow parameterq
Figure 7-2: Ion current vs flow rate. Formamide
5.4.2, from Lozano, [47]
100
+ NaI, dashed line corresponds to rq, Figure
j-FDM, f(e)491
1--4 0.85GO
4.10E-07 4-k- 0-85M__
pI._0825GO
3.10E-07
2.10E-07
1.10E-07
1.00E-08
-------1.5
-2
-----
----
---
----
2.5
,
3
TI
Figure 7-3: Current vs non-dimensional flow rate plot for formamide (K=0.01 Si/m). ho =
2pm. Two sets of results Go = 0.85 eV and 0.825eV, Iot is the total emitted current, Ii
is the accumulated ion current. FDM is the predicted current for the pure droplet regime,
Eq. 3.45.
(K = 2.9 Si/m). Note that the ion current is essentially zero with decreasing flow rate
until a threshold value is reached and then a very fast growth is seen, [47].
7.2
Modified solvation energies
Due to the numerical stiffness associated with running simulations at high conductivities
(and consequently low flow rates) we have been forced to limit ourselves to formamide
solutions with K = 0.01 Si/m. A consequence of the low conductivity of our simulation is
that the maximum E-field obtained is roughly one order of magnitude below the necessary
1 V/nm for field evaporation. For this reason our first approach to the ion evaporation
problem has been to lower the energy (Go) required to extract the ions from the liquid
surface. Preliminary results for these calculations are shown next.
Our initial results with a modified solvation energy predict, as expected, a higher total
current than that for the pure droplet regime. Notice however, that the ion current is an
increasing function of the flow rate which is at odds with the experimental evidence. Figure 7-3 has the non-dimensional flow rate 7 on the x-axis and the total emitted current
Itot as y-axis. Two sets of numerical results are shown, these correspond to two solvation
energies: Go = 0.85 eV and 0.825 eV respectively. For each set of results the total current
(Itot) and ion current (I,) are plotted as function of q. For reference the emitted current
predicted by Fernandez de la Mora and Loscertales for the pure droplet case is included
101
. . .
I
.
. .
I ...
A
0.8
V
-0.6
OA
jet radius h
0.2
00
0.05
0.1
0.05
0.1
0.15
0.2
0.25
0.3
0.35
OA
OAS
0.5
0.3
0.35
0.4
0A5
0.5
axial distanc
0.8.
V
0.6
0.4
0
0
0.15
0.2
0.25
axial ditac
Figure 7-4: Top plot : Depiction of jet radius (h), and ion current per unit length [3.57]
(A) vs axial position. Bottom plot: Electric currents vs axial position. I, jet convection
current (4.21), Icd jet conduction current (4.22), Ii accumulated ion current (3.58), Itot total
current
(FDM in the figure, Eq. 3.45). Notice that although the total current for both ion evaporation cases is higher than that for the pure droplet regime (in accordance with Gamero's
data), the observed ion current is still an important function of flow rate, which is at odds
with the results of Lozano, (Fig. 7-2, [47]).
In any case, some, important things can be learned from these results. From figure 7-4
we see the comparison of the jet shape along with the ion emitted current per unit length
(A).
These results have been normalized (y* = y(z)/max(y)) for each variable, and are
only meant to show the relative behavior of the important variables.
The maximum as
one would expect is located a bit downstream from the transition region (coinciding with
the maximum E, field). Other interesting effects can be seen from analyzing the current
profile of the jet.
The conduction current (Iem)
is still dominant in the cone section of
the jet with negligible contributions from the convection (I,.)
102
and accumulated ion current
0.9
0.8
0.7
0
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
z-axaS
0.6
0.7
0.8
0.9
1
Figure 7-5: Comparison of surface charge behavior for droplets and mixed-regime. Compare
to pure droplet cases, 6-13
(Ii). As the liquid traverses from the cone to the jet section the conduction current almost
disappears and now most of the current is transported by the convection current and the ion
current. The convection and ion current compete with each other since both draw from the
available free surface charge (o). If the ion evaporation is strong enough the ion current may
dominate and overcome the convection current. A second effect is that the surface charge
may develop a local depression at the location of maximum E-field (co-located with the
maximum evaporation section), see figure 7-5. The depression is created by the continuous
loss of ions (free charge) which cannot be replenished fast enough by the bulk of the liquid.
In any case, although these effects may be qualitatively correct they should be taken with
a grain of salt.
Another issue to which attention should be drawn too is that temperature may become
significant near (but not necessarily at) the maximum evaporation location. A calculation of
ohmic heating along the jet (see section 6.2.3) reveals that the point of maximum dissipation
is not co-located with the maximum ion evaporation point (and E-field), 7-6. This effect
may modify the ion evaporation region since the liquid temperature (T) does have an effect
on ion desorption. A heat transfer model of the temperature distribution along the jet, is
however, beyond the scope of this work. Nonetheless, it does mark a point to be considered
for future research.
A second attempt with the reduced solvation energy, but now following the technique
proposed by Higuera is presented next, [29]. Again, one can only expect to draw qualitative
103
0.9 0.8
I
a
current density for constant T
. -Ion
0.7 90.6
0.5
ohmic heating
- a
0.4
-
-
0.2-
0.1 ---0
0.1
0.2
0.3
0.4
05
z-axis
0.6
0.7
0.8
0.9
1
Figure 7-6: Normalized comparison of the ion current density (continuous line) and ohmic
heating distribution 0 = 21rK(Eth)2 /(1+ cos a), see section 6.2.3
conclusions from these simulations. The ion current density equation (3.59) is rewritten as:
=e
K
exp (A* [(E ) 1 / 2 _ (3)1/2])
(7.1)
where E, = E,/Eo is the nondimensional form of the normal electric field modified
to account for space charge effects (E,), (Eq. 3.62. E* is defined by equation 3.61 and
nondimensionalized as
E* = E*/Eo. The sensitivity parameter A* is then:
1/2
A* = (
(47reo
)
KT(7.2)
KBT
where the only new addition is a sensitivity constant c. There are various options for
the scaling E-field as described by Gamero,[21]. In this case we choose, for convenience:
Eo
= (
)
foRd)/
(7.3)
where Rd has been previously defined as the inertial length scale, Eq. 3.19. As mentioned
before (see chapter 3) ion evaporation becomes important once the maximum electric field
approaches Ema, = 1 V/nm, which for formamide (77 = 1) requires a conductivity is
K = 1.3 Si/m. The sensitivity parameter for a formamide solution under experimental
conditions (at room temperature (T = 300K), c = 1, and for an assumed Eo = 1 V/nm) is
A* = 46.34. Higuera quotes a value for A* of 112 for a formamide solution with K = 1 Si/m,
104
x10
x0
7
11
6
102
5
9
4
8
2500
1500
CU
13.
6
1000
2
5
1
.
0
0.5
z-direction [m]x
1 0
104
i'
500
~4
100'
0.5
1
z-direction [m] x 104
0
0.5
1
z-direction [m] x 10,
Figure 7-7: Results for ion evaporation using Higuera's method. Results for FM2 (K =
0.01 Si/m), Go = 0.8 eV, c = 0.85 for Na+ solvated ions. Plots show spatial evolution
relative to z-axis; left:jet radius, center: Normal external E-field, right: evaporated ion
current density. Arrows point towards lower flow rates (Q = 4,3,2) x 10-11M 3 /S
but with a calculated E 0 = 7 V/nm which may be a bit high, [29]. The discrepancy with
Higuera is due to the use of an alternative formula to estimate the maximum E-field.
In this case, the solvation energy Go is lowered as done before but now the A* parameter
is modified by a constant c. Basically, the factor c can be thought of as a modification of
the local temperature: Tm = T/.fr > T. This could be realistic depending on thermal
balances.
Again, these modifications can only hope to give us a qualitative feel for the
physics at work.
Simulations with this approach, have been performed for a formamide solution with
a conductivity (K = 0.01 Si/m), emanating from a conducting needle of inner radius
ho= 6/pm, operating at a voltage of V = 950 V with the ground electrode set at a distance of
L
=
150/im. The solvation energy has been lowered to Go = 0.8 eV (from the original 1.7 eV
i.e. 2.125 times), the A* parameter is lowered by a factor of c = 0.85, the desorbed ions are
singly charged and have the characteristics of solvated Na+ ions extracted experimentally
(m = 275 amu, q/m = 350, 000 C/Kg).
The results for three flow rates (Q = 4,3,2) x
10-11m 3 /s are seen in figure 7-7. The decreasing flow rates result in thinner jet radii as
expected, coupled with an increase in peak normal external field E0 which in turn results
in a higher ion current density, ji.
105
x10
4-
0.01[
-.
tot
3.5
3
0.008 *1
2.5-
Ji
0.006
- -
-I
2'I
0.004-
6
z-drcin
.
8
-
- -----
- -
-
I-
0
05 1.5
-
Icy
11
0.0020.5[
ci
2
4
6
z- direction [m]
10
8
x 10
Figure 7-8: Results for ion evaporation using Higuera's method. Plots show spatial evolution
relative to z-axis. Left: ion current per unit length; right: current plots: ion current I,
convection current Iv, conduction current Icd, and total current Itot = I+Iv+ Ic. Arrows
point towards lower flow rates (Q = 4,3,2) x 10-"1m 3 /s
The emitted currents for the same computational results are shown in figure 7-8. Notice
two opposing trends: first the jet radius (h) will become smaller for decreasing flow rates;
and second the ion evaporation current density (ji) increases for decreasing flow rates. The
relevance of this becomes apparent when one realizes that the ion current per unit length is
A = 27rjih. Our results show that for this artificially low Go and c parameter the jet radius
still dominates over the ion evaporation effect and therefore when integrated to obtain Ii it
gives the wrong trend, see figure 7-8. From these results, the conclusion is that the artificial
values for the solvation energy are too high, and therefore the results are what one would
expect. It may be that the values used for Go and c in these results may not be the necessary
ones to obtain a proper ion evaporation behavior.
Again, this feels like playing God, but taken with a grain of salt this results may point
to a qualitatively coherent picture of the onset of ion evaporation.
106
7.3
Summary
The first results obtained with the ion evaporation model showed some of the trends expected.
However, the current vs flow rate results were not physically correct.
The ion
current keeps decreasing as the flow rate decreases instead of growing as experimental data
shows. Although this results are not the best, at least the stage has been set for future
work on this subject.
107
Chapter 8
Conclusions and recommendations
Every work has its beginning and its end, which we are often reluctant to accept. Perhaps
this is so because endings often feel abrupt, a bit unnatural.
I believe that this applies
to research as well, how could a work end when so much still needs, cries, to be learned?
Perhaps the right way to think about this, so the cliche says, is that it is like running where
there really is no finish line, but only times when we stop to rest. There are always new
things to learn and for me, that is the beauty of research.
8.1
Conclusions
This work picked up where Prof.
Martinez-Sainchez and Vadim Khayms stopped and
through it all we have learned many things. If any, in the next few paragraphs we provide a short summary of those things we have learned. Some of them are related to the
physics of the problem, others to the mathematics and numerics of it, and a few others
are nothing but comments and suggestions that may, hopefully, make someone else's work
easier.
From the onset of the project I have seen this electrostatically accelerated jet as a subset
of the more general problem of free jets. The equations we use to describe the colloid thruster
are valid both for charged and uncharged jets. This point of view, is what originally drove
us to consider the inclusion of viscous effects in our equations even if for the most part the
electric current is a very weak function of it.
From the beginning of this project we have tried to compare and contrast our numerical
results with existing experimental data, and theoretical estimates.
108
Fortunately, in many
cases our results have compared favorably to others people's work. In what follows a short
list of some of the comparisons made and used to benchmark our results:
" Comparison to current and flow rate experimental data
" Comparison to theoretical estimates of current, liquid properties, and flow rate
" Comparison to analytical estimates of jet diameter, and e-field at the jet and cone
section
" Reproduced qualitative current vs Voltage behavior
" Analysis of transition length scales vs cone-jet transition radius, showed same trends
as experimental data
" Qualitative cone-angle behavior as function of voltage
" Electrode effects as secondary variable
" Weak viscous effect on current
This benchmarking process along with the analysis of our results is treated in detail in
chapter 6 and 7. In some cases, our results did not uncover new phenomena, but did confirm
what has been seen experimentally and from other simplified models. In other words, some
of the results we have obtained may have been hinted, or predicted via simplified models,
in those cases the value of our results is that without making the simplifying assumptions
we have reproduced that particular part of the cone-jet physics. What is valuable is that
this code may be used to explore the range of validity of some of the assumptions made
in simplified cases. This in turn allows us to gain a deeper understanding of the subject
matter.
8.2
Recommendations
All this is good, but one cannot underestimate the importance of the numerical side. For
this case, it may well be that we may have obtained as much mileage as we could from
the Runge-Kutta solver. A better approach, which would allow calculations for even stiffer
problems (i.e. low flow rates and high conductivities) would be a semi-implicit or implicit
109
solver. Said solver, would have to make use of an adaptive grid generator (already part of
this work.) The overhead of implicit and semi-implicit codes is higher, but in the event of
breakup may be the right choice.
Practical considerations are important, these are the ones that in the long run make our
life easier. The conical version of the equations we have used remains a more natural way to
describe the problem, but I believe that the simplicity of the cylindrical formulation constitutes a better compromise between accuracy and simplicity. Not to mention that cylindrical
version of the flow equations may be more amenable to higher order approximations than
the conical one.
The electrostatic part of the problem is currently solved using a panel method. This
is perhaps a good compromise for those cases where space charge is not a concern.
If
however, one wishes to explore those cases where this effects are important it would pay off
to seek alternatives. I think that pragmatism in this case is the answer. We are interested
in the physics, not the advancement of numerical schemes. Other works have used third
party Poisson solvers porting information between their code and the commercial code.
Sometimes it is not worthwhile to reinvent the wheel.
8.3
Future work
The ion evaporation section of this work marks only the first steps in this direction. Clearly,
the topic is rich and many things need to be done. First, and most important some means
for reaching the required E-field must be found (1 V/nm). A second approach, may be
through proper scaling of the solvation energy. At this point it is not clear, which path may
be easier.
Ohmic dissipation along the jet may have an impact on the temperature distribution of
the jet. Whether this will change dramatically the ion evaporation behavior remains to be
seen, but it should not be forgotten.
Jet breakup and drop formation Even though jet breakup is beyond the scope of this
work, we should point out that at least the some of the groundwork for this has been done.
I think that one of the more exciting aspects for future work in this code , or similar code
would be to address the jet breakup scenario. The information that could be obtained in
that case would be invaluable for plume codes as their upstream boundary condition. Such
110
a code would also be capable of exploring the physics of main and satellite droplets as well
as the nature of their charging. Another interesting aspect would be that a code that can
deal with breakup may explore the physics of ion emission from the breakup point, droplets,
and meniscus emission.
Bridging the gap between colloid and FEEP thrusters As mentioned by Lozano, this
may be one of the most interesting problems. The physics of FEEP's and colloid thrusters
are not that different. This problem would most likely benefit from a non-dimensional set
of equations, since scales will change dramatically. In our experience the solver (the RK4),
as would be expected, can cope only with some degree of stiffness. A rule of thumb may
be that the solver starts to have problems once the ratio of maximum to minimum radii
reaches a value of about 30. The message is, that designing a simulation that may bridge
this gap will require careful thought about the numerical schemes.
111
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